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A PRIMER OF REAL ANALYSIS

Dan Sloughter Furman University

Furman University A Primer of Real Analysis

Dan Sloughter

This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it is freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning.

The LibreTexts mission is to unite students, faculty and scholars in a cooperative effort to develop an easy-to-use online platform for the construction, customization, and dissemination of OER content to reduce the burdens of unreasonable textbook costs to our students and society. The LibreTexts project is a multi-institutional collaborative venture to develop the next generation of openaccess texts to improve postsecondary education at all levels of higher learning by developing an Open Access Resource environment. The project currently consists of 14 independently operating and interconnected libraries that are constantly being optimized by students, faculty, and outside experts to supplant conventional paper-based books. These free textbook alternatives are organized within a central environment that is both vertically (from advance to basic level) and horizontally (across different fields) integrated. The LibreTexts libraries are Powered by NICE CXOne and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. This material is based upon work supported by the National Science Foundation under Grant No. 1246120, 1525057, and 1413739. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation nor the US Department of Education. Have questions or comments? For information about adoptions or adaptions contact [email protected]. More information on our activities can be found via Facebook (https://facebook.com/Libretexts), Twitter (https://twitter.com/libretexts), or our blog (http://Blog.Libretexts.org). This text was compiled on 10/01/2023

TABLE OF CONTENTS Licensing

1: Fundamentals 1.1: Sets and Relations 1.2: Functions 1.3: Rational Numbers 1.4: Real Numbers

2: Sequences and Series 2.1: Sequences 2.2: Infinite Series

3: Cardinality 3.1: Binary Representations 3.2: Countable and Uncountable Sets 3.3: Power Sets

4: Topology of the Real Line 4.1: Intervals 4.2: Open Sets 4.3: Closed Sets 4.4: Compact Sets

5: Limits and Continuity 5.1: Limits 5.2: Monotonic Functions 5.3: Limits to Infinity and Infinite Limits 5.4: Continuous Functions

6: Derivatives 6.1: Best Linear Approximations 6.2: Derivatives 6.3: Mean Value Theorem 6.4: Discontinuities of Derivatives 6.5: L'Hopital's Rule 6.6: Taylor's Theorem

7: Integrals 7.1: Upper and Lower Integrals 7.2: Integrals 7.3: Integrability Conditions 7.4: Properties of Integrals 7.5: The Fundamental Theorem of Calculus 7.6: Taylor's Theorem Revisited

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7.7: An Improper Integral

8: More Functions 8.1: The Arctangent Function 8.2: The Tangent Function 8.3: The Sine and Cosine Functions 8.4: The Logarithm Functions 8.5: The Exponential Function

Index Glossary Detailed Licensing

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Licensing A detailed breakdown of this resource's licensing can be found in Back Matter/Detailed Licensing.

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CHAPTER OVERVIEW 1: Fundamentals 1.1: Sets and Relations 1.2: Functions 1.3: Rational Numbers 1.4: Real Numbers Thumbnail: The graph of f (x) = e has a tangent line with slope 1 at x = 0 . (CC BY; OpenStax) x

This page titled 1: Fundamentals is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1

1.1: Sets and Relations 1.1.1 The Integers Kronecker once said, "God made the integers; all the rest is the work of man." Taking this as our starting point, we assume the existence of the set Z = {… , −3, −2, −1, 0, 1, 2, 3, …}.

(1.1.1)

the set of integers. Moreover, we assume the properties of the operations of addition and multiplication of integers, along with other elementary properties such as the Fundamental Theorem of Arithmetic, that is, the statement that every integer may be factored into a product of prime numbers and this factorization is essentially unique.

1.1.2 Sets We will take a naive view of sets: given any property p, we may determine a set by collecting together all objects which have property p. This may be done either by explicit enumeration, such as, p is the property of being one of a, b, or c, which creates the set {a, b, c}, or by stating the desired property, such as, p is the property of being a positive integer, which creates the set +

Z

= {1, 2, 3, 4, …}.

(1.1.2)

The notation x ∈ A indicates that x is an element of the set A. Given sets A and B, we say A is a subset of B, denoted A ⊂ B, if from the fact that x ∈ A it necessarily follows that x ∈ B. We say sets A and B are equal if both A ⊂ B and B ⊂ A. Given two sets A and B, we call the set A ∪ B = {x : x ∈ A or x ∈ B}

(1.1.3)

A ∩ B = {x : x ∈ A and x ∈ B}

(1.1.4)

A∖B = {x : x ∈ A, x ∉ B}

(1.1.5)

the union of A and B and the set

the intersection of A and B. We call the set

the difference of A and B . More generally, if I is a set and {A

α

: α ∈ I}

is a collection of sets, one for each element of I , then we have the union ⋃ Aα = {x : x ∈ Aα for some α}

(1.1.6)

α∈I

and the intersection ⋂ Aα = {x : x ∈ Aα for all α} .

(1.1.7)

α∈I

Example 1.1.1 For example, if I

= {2, 3, 4, …}

and we let +

A2 = {n : n ∈ Z

+

, n > 1, n ≠ 2m for any m ∈ Z

with m > 1}

(1.1.8)

and, for any i ∈ I , i > 2 , +

Ai = {n : n ∈ Ai−1 , n ≠ mi for any m ∈ Z

then ∩

i∈I

Ai

with m > 1} ,

(1.1.9)

is the set of prime numbers.

If A and B are both sets, we call the set A × B = {(a, b) : a ∈ A, b ∈ B}

(1.1.10)

the cartesian product of A and B. If A = B, we write

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2

A

= A × A.

(1.1.11)

Example 1.1.2 2

Z

= {(m, n) : m ∈ Z, n ∈ Z}

is the set of all ordered pairs of integers.

1.1.3 Relations

Given two sets A and B, we call a subset R of A × B a relation. Given a relation R, we will write a ∼ is clear from the context, to indicate that (a, b) ∈ R.

R

b

, or simply

a ∼b

if R

Example 1.1.3 We say that an integer m divides and integer n if n = mi for some integer i. If we let R = {(m, n) : m ∈ Z, n ∈ Z, m divides n},

then R is a relation. For example, 3 ∼

R

12

(1.1.12)

.

Consider a set A and a relation R ⊂ A . For purposes of conciseness, we say simply that R is a relation on A. If R is such that a ∼ a for every a ∈ A, we say R is reflexive; if R is such that b ∼ Ra whenever a ∼ b, we say R is symmetric; if a ∼ b and b ∼ c together imply a ∼ c, we say R is transitive. We call a relation which is reflexive, symmetric, and transitive an equivalence relation. 2

R

R

R

R

R

Exercise 1.1.1 Show that the relation R on Z defined by m ∼

R

n

if m divides n is reflexive and transitive, but not symmetric.

n

if m − n is even is an equivalence relation.

Exercise 1.1.2 Show that the relation R on Z defined by m ∼

R

Given an equivalence relation R on a set A and an element x ∈ A, we call [x] = {y : y ∈ A, y ∼R x}

(1.1.13)

the equivalence class of x.

Exercise 1.1.3 Given an equivalence relation R on a set A, show that a. [x] ∩ [y] ≠ ∅ if and only if x ∼

R

b. [x] = [y] if and only if x ∼

R

y

y.

As a consequence of the previous exercise, the equivalence classes of an equivalence relation on a set A constitute a partition of A (that is, A may be written as the disjoint union of the equivalence classes).

Exercise 1.1.4 Find the equivalence classes for the equivalence relation in Exercise 1.1.2. This page titled 1.1: Sets and Relations is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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1.2: Functions If A and B are sets, we call a relation R ⊂ A × B a function with domain A if for every a ∈ A there exists one, and only one, b ∈ B such that (a, b) ∈ R. We typically indicate such a relation with the notation f : A → B, and write f (a) = b to indicate that (a, b) ∈ R. We call the set of all b ∈ B such that f (a) = b for some a ∈ A the range of f . With this notation, we often refer to R as the graph of f . We say f : A → B is one-to-one if for every b in the range of f there exists a unique a ∈ A such that f (a) = b. We say f is onto if for every b ∈ B there exists at least one a ∈ A such that f (a) = b. For example, the function f : Z → Z defined by f (z) = z is one-to-one, but not onto, whereas the function f : Z → Z defined by f (z) = z + 1 is both one-to-one and onto. +

+

2

Given two functions, g : A → B and f by f ∘ g(a) = f (g(a)) . If f

: B → C,

we define the composition, denoted

is both one-to-one and onto, then we may define a function f (a) = b . Note that this implies that f ∘ f (b) = b for all b ∈ B and f f. : A → B

−1

−1

f

−1

f ∘ g : A → C,

by requiring f (b) = a if and only if for all a ∈ A. We call f the inverse of

: B → A

∘ f (a) = a

to be the function defined −1

−1

Given any collection of nonempty sets, {A } , α ∈ I , we assume the existence of a function ϕ : I → B = ⋃ A , with the property that ϕ(α) ∈ A . We call such a function a choice function. The assumption that choice functions always exist is known as the Axiom of Choice. α

α∈I

α

α

This page titled 1.2: Functions is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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1.3: Rational Numbers Let P

= {(p, q) : p, q ∈ Z, q ≠ 0}.

We define an equivalence relation on P by saying (p, q) ∼ (s, t) if pt = qs .

Exercise 1.3.1 Show that the relation as just defined is indeed an equivalence relation. p

We will denote the equivalence class of (p, q) ∈ P by p/q, or . We call the set of all equivalence classes of numbers, which we denote by Q. If p ∈ Z, we will denote the equivalence class of (p, 1) by p; that is, we let q

P

the rational

p = p.

(1.3.1)

1

In this way, we may think of Z as a subset of Q.

1.3.1 Field Properties We wish to define operations of addition and multiplication on elements of Q. We begin by defining operations on the elements of P . Namely, given (p, q) ∈ P and (s, t) ∈ P , define (p, q) ⊕ (s, t) = (pt + sq, qt)

(1.3.2)

(p, q) ⊗ (s, t) = (ps, qt).

(1.3.3)

and

Now

suppose

and since

(p, q) ∼ (a, b)

(pt + sq, qt) ∼ (ad + cb, bd),

It

(s, t) ∼ (c, d).

follows

that

(p, q) ⊕ (s, t) ∼

(a, b) ⊕ (c, d),

that

(pt + sq)bd = pbtd + sdqb = qatd + tcqb = (ad + cb)qt.

is,

(1.3.4)

Moreover, (p, q) ⊗ (s, t) ∼ (a, b) ⊗ (c, d), that is, (ps, qt) ∼ (ac, bd), since psbd = pbsd = qatc = qtac.

(1.3.5)

This shows that the equivalence class of a sum or product depends only on the equivalence classes of the elements being added or multiplied. Thus we may define addition and multiplication on Q by p

s

pt + sq

+ q

=

(1.3.6)

t

qt

and p

s ×

q

ps =

t

,

(1.3.7)

qt

and the results will not depend on which representatives we choose for each equivalence class. Of course, multiplication is often denoted using juxtaposition, that is, p

s ×

q

p s =

t

,

and repeated multiplication may be denoted by exponentiation, that is, itself n times. Note that if (p, q) ∈ P , then (−p, q) ∼ (p, −q). Hence, if a =

p q

∈ Q,

−p −a =

(1.3.8)

q t n

a , a ∈ Q

and

+

n ∈ Z

,

represents the product of

a

with

then we let p

= q

.

(1.3.9)

−q

For any a, b ∈ Q, we will write a − b to denote a + (−b) . If a =

p q

∈ Q

with p ≠ 0, then we let

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−1

a

q =

.

(1.3.10)

,

(1.3.11)

p

Moreover, we will write 1

−1

=a a 1

−n

n

=a

(1.3.12)

a

for any n ∈ Z

+

,

and, for any b ∈ Q , b

−1

= ba

.

(1.3.13)

a

It is now easy to show that 1. a + b = b + a for all a, b ∈ Q ; 2. (a + b) + c = a + (b + c)

for all a, b, c ∈ Q;

3. ab = ba for all a, b ∈ Q ; 4. (ab)c = a(bc) for all a, b, c ∈ Q; 5. a(b + c) = ab + ac for all a, b, c ∈ Q; 6. a + 0 = a for all a ∈ Q ; 7. a + (−a) = 0 for all a ∈ Q ; 8. 1a = a for all a ∈ Q ; 9. if a ∈ Q, a ≠ 0, then aa

−1

=1

.

Taken together, these statements imply that Q is a field.

1.3.2 Order and Metric Properties We say a rational number a is positive if there exist p, q ∈ Z such that a = +

+

Q

p q

. We denote the set of all positive elements of Q by

.

Given a, b ∈ Q, we say a is less than b, or, equivalently, b is greater than a, denoted either by a < b or b > a , if b − a is positive. In particular, a > 0 if and only if a is positive. If a < 0, we say a is negative. We write a ≤ b, or, equivalently, b ≥ a, if either a < b or a = b .

Exercise 1.3.2 Show that for any a ∈ Q, one and only one of the following must hold: (a) a < 0, (b)a = 0, (c)a > 0 .

Exercise 1.3.3 Show that if a, b ∈ Q

+

,

then a + b ∈ Q . +

Exercise 1.3.4 Suppose a, b, c ∈ Q. Show each of the following: a. One, and only one, of the following must hold: (i) a < b , (ii) a = b , (iii) a > b . b. If a < b and b < c, then a < c .

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c. If a < b, then a + c < b + c . d. If a > 0 and b > 0, then ab > 0 .

Exercise 1.3.5 Show that if a, b ∈ Q with a > 0 and b < 0, then ab < 0 .

Exercise 1.3.6 Show that if a, b, c ∈ Q with a < b, then ac < bc if c > 0 and ac > bc if c < 0 .

Exercise 1.3.7 Show that if a, b ∈ Q with a < b, then a+b a

0, it follows that na − b > 0, that is, na > b. We say that the ordered field Q is archimedean. Note that it also follows that we may choose n large enough to ensure that < a . b

n

1.3.3 Upper and Lower Bounds

Definition Let A ⊂ Q. If s ∈ Q is such that s ≥ a for every a ∈ A, then we call s an upper bound for A . If s is an upper bound for A with the property that s ≤ t whenever t is an upper bound for A, then we call s the supremum, or least upper bound, of A, denoted s = sup A. Similarly, if r ∈ Q is such that r ≤ a for every a ∈ A, then we call r a lower bound for A. If r is a lower bound for A with the property that r ≥ t whenever t is a lower bound for A, then we call r the infimum, or greatest lower bound, of A, denoted r = inf A.

Exercise 1.3.9 Show that the supremum of a set definition.

A ⊂ Q,

if it exists, is unique, and thus justify the use of the definite article in the previous

A set which does not have an upper bound will not, need not have a supremum.

fortiori, have a supremum. Moreover, even sets which have upper bounds

a

Example 1.3.1 Q

does not have an upper bound.

Example 1.3.2 Consider the set +

A = {a : a ∈ Q

2

,a

< 2} .

(1.3.20)

Note that if a, b ∈ Q with a < b, then +

2

b

from which it follows that a bound for A.

2

Now suppose s ∈ Q

+

Suppose s

2

2,

then a is an upper bound for A. For example, 4 is an upper

is the supremum of A. We must have either s

2

and let ϵ = 2 − s

2

.

(1.3.21)

2

< 2, s

> 2,

or s

2

=2

.

By the archimedean property of Q, we may choose n ∈ Z

+

such that

2s + 1 < ϵ,

(1.3.22)

n

from which it follows that 2s n

2s +

1 +

2

n

=

1 n

n

2s + 1 ≤

< ϵ.

(1.3.23)

n

Hence

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2

1 (s +

2

)

=s

2s

which implies that s + So now suppose s

2

1 n

> 2.

since s < s +

∈ A.

Again let n ∈ Z

1 n

1

+

+

n

2 2

)

=s

2s −

n − 2,

+ ϵ = 2,

(1.3.24)

this contradicts the assumption that s is an upper bound for A.

,

1

2

s

1 −ϵ+

n

Thus s −

1 n

is an upper bound for A and s −

1 n

2

1 =2+

n

< s,

2

> 2.

(1.3.27)

n

contradicting the assumption that s = sup A .

Thus we must have s = 2. However, this is impossible in light of the following proposition. Hence we must conclude that A does not have a supremum. 2

Proposition 1.3.2 There does not exist a rational number s with the property that s

2

=2

.

Proof Suppose there exists s ∈ Q such that s factor other than 1 in common) and s =

2

Choose a, b ∈ Z Then

+

= 2. a b

.

so that

a

and b are relatively prime (that is, they have no

2

a

= 2,

(1.3.28)

2

b

so a

2

2

= 2b .

Thus a , and hence a, is an even integer. So there exists c ∈ Z such that a = 2c. Hence +

2

2

2b

from which it follows that b relatively prime. Q.E.D.

2

= 2c,

and so

b

2

=a

2

= 4c ,

(1.3.29)

is also an even integer. But this contradicts the assumption that

a

and

b

are

Exercise 1.3.10 Show that there does not exist a rational number s with the property that s

2

=3

.

=6

.

Exercise 1.3.11 Show that there does not exist a rational number s with the property that s

2

Exercise 1.3.12 Let A = {a : a ∈ Q, a

3

< 2}

.

1. Show that if a ∈ A and b < a, then b ∈ A . 2. Show that if a ∉ A, and b > a, then b ∉ A . 1.3.4 Sequences

1.3.5

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Definition Suppose n ∈ Z, I

and A is a set. We call a function φ : I

= {n, n + 1, n + 2, …},

→ A

a sequence with values in A .

Frequently, we will define a sequence φ by specifying its values with notation such as, for example, {φ(i)} , or {φ(i)} . Thus, for example, {i } denotes the sequence φ : Z → Z defined by φ(i) = i . Moreover, it is customary to denote the values of a sequence using subscript notation. Thus if a = φ(i), i ∈ I , then {a } denotes the sequence φ. For example, we may define the sequence of the previous example by writing a = i , i = 1, 2, 3, … . ∞

i∈I

2

∞

+

i=n

2

i=1

i

i

i∈I

2

i

Definition Suppose {a } there exists N i

is a sequence with values in Q. We say that {a ∈ Z such that

i }i∈I

i∈I

converges, and has limit L, L ∈ Q, if for every ϵ ∈ Q

+

| ai − L| < ϵ whenever i > N .

If the sequence {a

i }i∈I

,

(1.3.30)

converges to L, we write lim ai = L.

(1.3.31)

i→∞

Example 1.3.3 We have 1 lim i→∞

= 0,

(1.3.32)

i

since, for any rational number ϵ > 0 , 1 ∣1 ∣ ∣ − 0∣ = N , where N is any integer larger than

1 ϵ

(1.3.33)

.

Definition Suppose {a such that

i }i∈I

is a sequence with values in Q. We call {a

i }i∈I

a Cauchy sequence if for every ϵ ∈ Q

+

| ai − ak | < ϵ whenever both i > N and k > N .

,

there exists

N ∈ Z

(1.3.34)

Proposition 1.3.3 If {a

i }i∈I

converges, then {a

i }i∈I

is a Cauchy sequence.

Proof Suppose lim

i→∞

ai = L.

Given ϵ ∈ Q

+

,

choose an integer N such that ϵ | ai − L|

N . Then for any i, k > N , we have ϵ | ai − ak | = |(ai − L) + (L − ak )| ≤ | ai − L| + | ak − L|

0,

and

aN < xi < bN

(1.3.42)

for all i > N . Moreover, 1 | bN − aN | =

N −1

,

(1.3.43)

2

so 1 | xi − xk |

N . Hence given any ϵ ∈ Q

+

,

if we choose an integer N such that 2

N −1

>

1 ϵ

,

then

1 | xi − xk |

0. But if f (s) > 0, then there exists t ∈ Q such that t < s and f (t) > 0. We can then choose N so that t < a , implying that ∞

N

N

i

i=1

+

N

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contradicting the construction of {a } . Hence we must have Thus we must conclude that {x } does not converge. ∞

f (aN ) > 0,

i

i=1

f (s) = 0,

which is not possible since

s ∈ Q

.

∞

i

i=1

This page titled 1.3: Rational Numbers is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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1.4: Real Numbers Let C be the set of all Cauchy sequences of rational numbers. We define a relation on C as follows: If {a } and {b } are Cauchy sequences in Q, then {a } ∼ {b } , which we will write more simply as a ∼ b , if for every rational number ϵ > 0, there exists an integer N such that i

i

i∈I

j

i

j∈J

j

i∈I

j∈J

i

| ai − bi | < ϵ

(1.4.1)

whenever i > N . This relation is clearly reflexive and symmetric. To show that it is also transitive, and hence an equivalence relation, suppose a ∼ b and b ∼ c . Given ϵ ∈ Q , choose N so that +

i

i

i

i

ϵ | ai − bi |

N and M so that ϵ | bi − ci |

M . Let L be the larger of N and M . Then, for all i > L , ϵ | ai − ci | ≤ | ai − bi | + | bi − ci |

0, choose integers N and M such that

i }i∈I

k

j

k

k }k∈K

j∈J

k

ϵ | ai − aj |

N and ϵ | bi − bj |

M . If L is the larger of N and M , then, for all i, j > L , ϵ | si − sj | = |(ai − aj ) + (bi − bj )| ≤ | ai − aj | + | bi − bj |

L , ϵ |(ai + bi ) − (ci + di )| ≤ | ai − ci | + | bi − di |

0 choose integers N and M such that

i }i∈I

j }j∈J

K =I ∩J

i∈I

i

i∈K

are both Cauchy sequences of rational numbers. k

k

k∈K

k

k

B >0

be an upper bound for the set

{| ai | : i ∈ I } ∪ {| bj | : j ∈ J} .

ϵ | ai − aj |

N and ϵ | bi − bj |

M . If L is the larger of N and M , then, for all i, j > L , | pi − pj | = | ai bi − aj bj | = | ai bi − aj bi + aj bi − aj bj | = | bi (ai − aj ) + aj (bi − bj )| ≤ | bi (ai − aj )| + | aj (bi − bj )| = | bi | | ai − aj | + | aj | | bi − bj | ϵ

ϵ

0, choose integers N and M such that

{ di }

{| bj | : j ∈ J} ∪ {| ci | : i ∈ H }

i

i∈G

i

i

i.

Let

B >0

be an upper bound for the set

ϵ | ai − ci |

N and ϵ | bi − di |

M . If L is the larger of N and M , then, for all i > L , | ai bi − ci di |

= | ai bi − bi ci + bi ci − ci di | = | bi (ai − ci ) + ci (bi − di )| ≤ | bi (ai − ci )| + | ci (bi − di )| = | bi | | ai − ci | + | ci | | bi − di | ϵ 0 and an integer N such that | a | > δ for all i > N . It should be clear that any sequence which converges to 0 is not bounded away from 0. Moreover, as a i

i∈I

i

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consequence of the next exercise, any Cauchy sequence which does not converge to 0 must be bounded away from 0.

Exercise 1.4.2 Suppose

is a Cauchy sequence which is not bounded away from 0. Show that the sequence converges and

{ ai }i∈I

.

limi→∞ ai = 0

Exercise 1.4.3 Suppose {a from 0.

i }i∈I

is a Cauchy sequence which is bounded away from 0 and a

i

∼ bi .

Show that

Now suppose {a } is a Cauchy sequence which is bounded away from 0 and choose i > N . Define a new sequence { c } + 1 by setting i

i∈I

δ >0

{ bj }

j∈J

and

N

is also bounded away

so that

| ai | > δ

for all

∞

i

i=N +1

1 ci =

, i = N + 1, N + 2, …

(1.4.16)

ai

Given ϵ > 0, choose M so that | ai − aj | < ϵδ

2

(1.4.17)

for all i, j > M . Let L be the larger of N and M . Then, for all i, j > L, we have ∣ 1 | ci − cj | = ∣

−

∣ ai

1 ∣ ∣ aj ∣

∣ aj − ai ∣ =∣ ∣

ai aj

∣ ∣

| aj − ai | = | ai aj | ϵδ < δ

2

2

= ϵ.

Hence {c

∞

i }i=N +1

is a Cauchy sequence.

Now suppose {b } is a Cauchy sequence with a ∼ b . By Exercise 1.4.3 we know that {b so choose γ > 0 and K such that |b | > γ for all j > K. Given ϵ > 0, choose P so that j

i

j∈J

j }j∈J

i

is also bounded away from

0,

j

| ai − bi | < ϵδγ.

(1.4.18)

for all i > P . Let S be the larger of N , K, and P . Then, for all i, j > S, we have 1 ∣ ∣ 1 ∣ bi − ai ∣ ∣ − ∣ =∣ ∣ ∣ ai ∣ ∣ bi ∣ ai bi =

| bi − ai | | ai bi | ϵδγ

< δγ = ϵ.

Hence ∼ . Thus if then we may define 1

1

ai

bi

u ≠0

is a real number which is the equivalence class of

−1

a

{ ai }i∈I ( necessarily bounded away from 0),

1 =

(1.4.19) a

to be the equivalence class of

1.4.3

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∞

1 {

} ai

where N has been chosen so that |a

i|

>δ

,

(1.4.20)

i=N +1

for all i > N and some δ > 0.

It follows immediately from these definitions that R is a field. That is: 1. a + b = b + a for all a, b ∈ R ; 2. (a + b) + c = a + (b + c)

for all a, b, c ∈ R;

3. ab = ba for all a, b ∈ R ; 4. (ab)c = a(bc) for all a, b, c ∈ R; 5. a(b + c) = ab + ac for all a, b, c ∈ R; 6. a + 0 = a for all a ∈ R ; 7. a + (−a) = 0 for all a ∈ R ; 8. 1a = a for all a ∈ R ; 9. if a ∈ R, a ≠ 0, then aa

−1

=1

.

1.4.2 Order and Metric Properties Definition Given u ∈ R, we say that u is positive, written u > 0, if u is the equivalence class of a Cauchy sequence {a } for which there exists a rational number ϵ > 0 and an integer N such that a > ϵ for every i > N . A real number u ∈ R is said to be negative if −u > 0. We let R denote the set of all positive real numbers. i

i∈I

i

+

Exercise 1.4.4 Show that if u ∈ R, then one and only one of the following is true: (a)u > 0, (b)u < 0, or (c)u = 0 .

Exercise 1.4.5 Show that if a, b ∈ R

+

,

then a + b ∈ R . +

Definition Given real numbers u and v, we say u is greater than v , written u > v, or, equivalently, v is less than u, written, v < u, if u − v > 0. We write u ≥ v, or, equivalently, v ≤ u, to indicate that u is either greater than or equal to v. We say that u is nonnegative if u ≥ 0 .

Exercise 1.4.6 Show that R is an ordered field, that is, verify the following: a. For any a, b ∈ R, one and only one of the following must hold: (i)a < b, (ii) a = b, ( iii) a > b . b. If a, b, c ∈ R with a < b and b < c, then a < c . c. If a, b, c ∈ R with a < b, then a + c < b + c . d. If a, b ∈ R with a > 0 and b > 0, then ab > 0 .

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Exercise 1.4.7 Show that if a, b ∈ R with a > 0 and b < 0, then ab < 0 .

Exercise 1.4.8 Show that if a, b, c ∈ R with a < b, then ac < bc if c > 0 and ac > bc if c < 0 .

Exercise 1.4.9 Show that if a, b ∈ R with a < b, then for any real number λ with 0 < λ < 1, a < λa + (1 − λ)b < b .

Definition For any a ∈ R, we call |a| = {

a,

if a ≥ 0,

−a,

if a < 0,

(1.4.21)

the absolute value of a .

Exercise 1.4.10 Show that for any a ∈ R, −|a| ≤ a ≤ |a| .

Proposition 1.4.1 For any a, b ∈ R, |a + b| ≤ |a| + |b| . Proof If a + b ≥ 0, then |a| + |b| − |a + b| = |a| + |b| − a − b = (|a| − a) + (|b| − b).

Both of the terms on the right are nonnegative by Exercise follows. If a + b < 0, then

1.4.10.

(1.4.22)

Hence the sum is nonnegative and the proposition

|a| + |b| − |a + b| = |a| + |b| + a + b = (|a| + a) + (|b| + b).

(1.4.23)

Again, both of the terms on the right are nonnegative by Exercise 1.4.10.Hence the sum is nonnegative and the proposition follows. Q. E. D. It is now easy to show that the absolute value function satisfies 1. |a − b| ≥ 0 for all a, b ∈ R, with |a − b| = 0 if and only if a = b, 2. |a − b| = |b − a| for all a, b ∈ R , 3. |a − b| ≤ |a − c| + |c − b| for all a, b, c ∈ R. These properties show that the function d(a, b) = |a − b|

(1.4.24)

is a metric, and we will call |a − b| the distance from a to b .

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Proposition 1.4.2 Given a ∈ R

+

,

there exist r, s ∈ Q such that 0 < r < a < s .

Proof Let {u} be a Cauchy sequence in the equivalence class of a. Since a > 0, there exists a rational ϵ > 0 and an integer N such that u > ϵ for all i > N . Let r = . Then u − r > for every i > N , so a − r > 0, that is, 0 < r < a. i∈I

ϵ

i

ϵ

i

2

Now choose an integer M so that |u

i

− uj | < 1

2

for all i, j > M . Let s = u

M+1

+ 2.

Then

s − ui = uM+1 + 2 − ui > 1

for all i > M . Hence s > a .

(1.4.25)

Q.E.D.

Proposition 1.4.3 R

is an archimedean ordered field.

Proof Given real numbers a and b with 0 < a < b, let r and s be rational numbers for which an archimedean field, there exists an integer n such that nr > s. Hence na > nr > s > b.

0 < r < a < b < s.

since Q is a

(1.4.26)

Q.E.D.

Proposition 1.4.4 Given a, b ∈ R with a < b, there exists r ∈ Q such that a < r < b . Proof Let

be a Cauchy sequence in the equivalence class of a and let {v} be in the equivalence class of b. Since there exists a rational ϵ > 0 and an integer N such that v − u > ϵ for all i > N . Now choose an integer M −u | < for all i, j > M . Let r = u + . Then

{u}i∈I

b − a > 0,

so that |u

i

j∈J

i

e

j

4

i

ϵ

M+1

2

ϵ r − ui

= uM+1 +

2

− ui

ϵ = 2

− (ui − uM+1 )

ϵ >

ϵ −

2

4

ϵ = 4

for all i > M and ϵ vi − r = vi − uM+1 − 2 ϵ = (vi − ui ) − (uM+1 − ui ) − ϵ > ϵ−

2

ϵ −

4

2

ϵ = 4

for all i larger than the larger of N and M . Hence a < r < b.

1.4.6

Q.E.D.

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1.4.3 Upper and Lower Bounds

Definition Let A ⊂ R . If s ∈ R is such that s ≥ a for every a ∈ A, then we call s an upper bound for A . If s is an upper bound for A with the property that s ≤ t whenever t is an upper bound for A, then we call s the supremum, or least upper bound, of A, denoted s = sup A . Similarly, if r ∈ R is such that r ≤ a for every a ∈ A, then we call r a lower bound for A. If r is a lower bound for A with the property that r ≥ t whenever t is a lower bound for A, then we call r the infimum, or greatest lower bound, of A, denoted r = inf A.

Theorem 1.4.5 Suppose A ⊂ R, A ≠ ∅, has an upper bound. Then sup A exists. Proof Let a ∈ A and let For i > 1, let

b

be an upper bound for

A.

Define sequences

and {b

∞

{ ai }

∞

i }i=1

i=1

as follows: Let

a1 = a

and

b1 = b.

ai−1 + bi−1 c =

.

(1.4.27)

2

If c is an upper bound for A, let a

i

= ai−1

and let b

i

otherwise, let a

= ci

i

=c

and b

i

Then

= bi−1 .

|b − a| | bi − ai | =

(1.4.28)

i−1

2

for i = 1, 2, 3, … Now, for choose N so that

i = 1, 2, 3, … ,

let r be a rational number such that a i

i

< ri < bi .

Given any

we may

ϵ > 0,

|b − a|

N

2

>

.

(1.4.29)

ϵ

Then, whenever i > N and j > N , |b − a| | ri − rj | < | bN +1 − aN +1 | =

N

< ϵ.

(1.4.30)

2

Hence

is a Cauchy sequence. Let i = 1, 2, 3, … , a ≤ s ≤ b . ∞

{ ri }

i=1

i

be the equivalence class of

s ∈ R

∞

{ ri }

i=1

.

Note

that,

for

i

Now if s is not an upper bound for that

A,

then there exists

N

2

a ∈ A

with a > s. Let

δ = a−s

and choose an integer

N

such

|b − a| >

.

(1.4.31)

δ

Then |b − a| bN +1 ≤ s +

< s + δ = a.

(1.4.32)

N

2

But, by construction, b

N +1

is an upper bound for A. Thus s must be an upper bound for A.

Now suppose t is another upper bound for A and t < s. Let δ = s − t and choose an integer N such that N

2

|b − a| >

.

(1.4.33)

δ

Then |b − a| aN +1 ≥ s −

> s − δ = t,

(1.4.34)

N

2

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which implies that a is an upper bound for A. But, by construction, a be the least upper bound for A, that is, s = sup A. Q.E.D. N +1

N +1

is not an upper bound for

A

. Hence s must

Exercise 1.4.11 Show that if

is nonempty and has a lower bound, then inf where −A = {x : −x ∈ A}).

A ⊂R

A = − sup(−A),

A

exists. (Hint: You may wish to first show that inf

This page titled 1.4: Real Numbers is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.8

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CHAPTER OVERVIEW 2: Sequences and Series 2.1: Sequences 2.2: Infinite Series Thumbnail: For the alternating harmonic series, the odd terms S in the sequence of partial sums are decreasing and bounded below. The even terms S are increasing and bounded above. (CC BY-SA-NC; OpenStax) 2k+1

2k

This page titled 2: Sequences and Series is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1

2.1: Sequences Definition Let {a } be a sequence of real numbers. We say exists an integer N such that i

converges, and has limit

{ ai }

i∈I

i∈I

L,

if for every real number

| ai − L| < ϵ whenever i > N .

We say a sequence {a

ϵ>0

there

(2.1.1)

which does not converge diverges.

i }i∈I

Definitions: nondecreasing and nonincreasing sequences We say a sequence {a We say a sequence {a

i }i∈I i}

i∈I

is nondecreasing if a is nonincreasing if a

i+1

i+1

≥ ai ≤ ai

for every i ∈ I and increasing if a for every i ∈ I and decreasing if a

i+1

> ai

i+1

< ai

for every i ∈ I . for every i ∈ I .

Definition: bounded sequences We say a set A ⊂ R is bounded if there exists a real number M such that |a| ≤ M for every a ∈ A. We say a sequence {a } of real numbers is bounded if there exists a real number M such that |a | ≤ M for all i ∈ I . i

i

i∈I

Theorem 2.1.1 If {a

i }i∈I

is a nondecreasing, bounded sequence of real numbers, then {a

i }i∈I

converges.

Proof Since {a } is bounded, the set of A = {a : i ∈ I } has a supremum. Let L = sup A. For any ϵ > 0, there must exist N ∈ I such that a > L − ϵ (or else L − ϵ would be an upper bound for A which is smaller than L ). But then i

i

i∈I

N

L − ϵ < aN ≤ ai ≤ L < L + ϵ

(2.1.2)

| ai − L| < ϵ

(2.1.3)

L = lim ai .

(2.1.4)

for all i ≥ N , that is,

for all i ≥ N . Thus {a

i }i∈I

converges and i→∞

Q.E.D. We conclude from the previous theorem that every nondecreasing sequence of real numbers either has a limit or is not bounded, that is, is unbounded.

Exercise 2.1.1 Show that a nonincreasing, bounded sequence of real numbers must converge.

Definition Let {a

be a sequence of real numbers. If for every real number M there exists an integer N such that a then we say the sequence {a } diverges to positive infinity, denoted by

i }i∈I

i > N,

i

i

>M

whenever

i∈I

lim ai = +∞.

(2.1.5)

i→∞

Similarly, if for every real number M there exists an integer N such that a {a } diverges to negative infinity, denoted by

i

i

N , then we say the sequence

i∈I

2.1.1

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lim ai = −∞.

(2.1.6)

i→∞

Exercise 2.1.2 Show that a nondecreasing sequence of real numbers either converges or diverges to positive infinity.

Exercise 2.1.3 Show that a nonincreasing sequence of real numbers either converges or diverges to negative infinity.

2.1.1 Extended Real Numbers It is convenient to add the symbols +∞ and −∞ to the real numbers R. Although neither +∞ nor −∞ is a real number, we agree to the following operational conventions: 1. Given any real number r, −∞ < r < ∞ . 2. For any real number r, r + (+∞) = r − (−∞) = r + ∞ = +∞,

(2.1.7)

r + (−∞) = r − (+∞) = r − ∞ = −∞,

(2.1.8)

and r

r =

+∞

= 0.

(2.1.9)

−∞

3. For any real number r > 0, r ⋅ (+∞) = +∞ and r ⋅ (−∞) = −∞ . 4. For any real number r < 0, r ⋅ (+∞) = −∞ and r ⋅ (−∞) = +∞ . 5. If a

= −∞, i = 1, 2, 3, … ,

then lim

ai = −∞

.

6. If a

= +∞, i = 1, 2, 3, … ,

then lim

ai = +∞

.

i

i

i→∞

i→∞

Note that with the order relation defined in this manner, +∞ is an upper bound and −∞ is a lower bound for any set A ⊂ R . Thus if A ⊂ R does not have a finite upper bound, then similarly, if A ⊂ R does not have a finite lower bound, then inf A = −∞ .

sup A = +∞;

When working with extended real numbers, we refer to the elements of R as finite real numbers and the elements +∞ and −∞ as infinite real numbers.

Exercise 2.1.4 Do the extended real numbers form a field?

2.1.2 Limit Superior and Inferior Definition Let {a

i }i∈I

be a sequence of real numbers and, for each i ∈ I , let u

i

= sup { aj : j ≥ i} .

lim sup ai = +∞;

If u

i

= +∞

for every i ∈ I , we let (2.1.10)

i→∞

otherwise, we let lim sup ai = inf { ui : i ∈ I } .

(2.1.11)

i→∞

In either case, we call lim inf

n→∞

an

the limit inferior of the sequence {a

i }i∈I

2.1.2

.

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Exercise 2.1.5 Given a sequence {a

i }i∈I

,

define {u

i }i∈I

and {l

i }i∈I

as in the previous two definitions. Show that

lim sup ai = lim ui

(2.1.12)

i→∞

i→∞

and lim inf ai = lim li . i→∞

(2.1.13)

i→∞

Exercise 2.1.6 Find lim sup a. a

i→∞

ai

and lim inf

i→∞

ai

∞

for the sequences {a

i }i=1

as de fined below.

i

i

= (−1 )

b. a

i

=i

c. a

i

=2

−i

d. a

i

=

1 i

The following proposition is often called the squeeze theorem.

Proposition 2.1.2 Suppose

{ ai }

ai ≤ ci ≤ bi

i∈I

, { bj }

j∈J

,

and

{ ck }

k∈K

are sequences of real numbers for which there exists an integer

N

such that

whenever i > N . If lim ai = lim bi ,

(2.1.14)

lim ci = lim ai = lim bi .

(2.1.15)

i→∞

i→∞

then i→∞

i→∞

i→∞

Proof Let L = lim

i→∞

ai = limi→∞ bi .

Suppose L is finite. Given ϵ > 0, there exists an integer M such that ϵ | ai − L|

N.

Show

N

such that

ci ≤ bi

whenever

i > N.

Show

i

Exercise 2.1.8 Suppose {b that if lim

j} j∈J

i→∞

and

are sequences for which there exists an integer then lim c = −∞.

{ ck }

k∈K

bi = −∞,

i→∞

i

Exercise 2.1.9 Suppose {a } and {b } are sequences of real numbers with a sequences converge, show that i

j

i∈I

i

j∈J

≤ bi

for all i larger than some integer N . Assuming both

lim ai ≤ lim bi .

(2.1.20)

lim inf ai ≤ lim sup ni .

(2.1.21)

i→∞

i→∞

Exercise 2.1.10 Show that for any sequence {a

i }i∈I

, i→∞

i→∞

Proposition 2.1.3 Suppose {a

i }i∈I

is a sequence for which lim sup ai = lim inf ai .

(2.1.22)

i→∞

i→∞

Then lim ai = lim sup ai = lim inf ai .

i→∞

(2.1.23)

i→∞

i→∞

Proof Let u

i

= sup { ak : k ≥ i}

and l

i

= inf { ak : k ≥ i} .

Then l

i

≤ ai ≤ ui

for all i ∈ I . Now

lim li = lim inf ai = lim sup ai = lim ui , i→∞

i→∞

so the result follows from the squeeze theorem.

i→∞

(2.1.24)

i→∞

Q.E.D.

Exercise 2.1.11 Suppose u is a real number such that u ≥ 0 and u < ϵ for any real number ϵ > 0. Show that u = 0 . 2.1.3 Completeness

Definition Suppose {a

i }i∈I

is a sequence in R. We call {a

i }i∈I

a Cauchy sequence if for every ϵ > 0 there exists an integer N such that | ai − aj | < ϵ

(2.1.25)

whenever both i > N and j > N .

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Theorem 2.1.4 Suppose {a

is a Cauchy sequence in R. Then {a

i }i∈I

i }i∈I

converges to a limit L ∈ R .

Proof Let u

i

and l = inf {a : k ≥ i} . Given any ϵ > 0 , there exists N Thus, for all i, j > N , a < a + ϵ and so

= sup { ak : k ≥ i}

i, j > N .

i

∈ Z

k

i

j

such that |a

i

− aj | < ϵ

ai ≤ inf { aj + ϵ : j ≥ i} = li + ϵ

for all i > N . Since {l

i }i∈I

for all

1

(2.1.26)

is a nondecreasing sequence, ai ≤ sup { li + ϵ : i ∈ I } = lim inf ai + ϵ

(2.1.27)

ui = sup { ak : k ≥ i} ≤ lim inf ai + ϵ

(2.1.28)

lim sup ai = inf { ui : i ∈ I } ≤ lim inf ai + ϵ.

(2.1.29)

i→∞

for all i > N . Hence i→∞

for all i > N . Thus i→∞

i→∞

Since lim inf

i→∞

ai ≤ lim sup i→∞ ai ,

it follows that ∣ ∣ ∣lim sup ai − lim inf ai ∣ ≤ ϵ. ∣ i→∞ i→∞ ∣

since this is true for every ϵ > 0, we have lim sup Q.E.D.

i→∞

(2.1.30)

and so {a

ai = limi→∞ inf ai ,

i }i∈I

converges by Proposition 2.1.3.

As a consequence of the previous theorem, we say that R is a complete metric space.

Exercise 2.1.12 Suppose A ⊂ R, A ≠ ∅, and s = sup A. Show that there exists a sequence {a

∞

i }i=1

with a

i

∈ A

such that lim

i→∞

ai = s

.

Exercise 2.1.13 Given a real number x ≥ 0, show that there exists a real number s ≥ 0 such that s

2

=x

.

−

We let √x denote the number s in the previous exercise, the square root of x. 2.1.4 Some Basic Results About Sequences

Proposition 2.1.5 Suppose {x } oonverges and i

i∈I

is a convergent sequence in

R, α

is a real number, and

L = limi→∞ xi .

Then the sequence

lim α xi = αL.

{α xi }i∈I

(2.1.31)

i→∞

Proof If α = 0, then {α x

i }i∈I

clearly converges to 0. So assume α ≠ 0. Given ϵ > 0, choose an integer N such that ϵ | xi − L|

N . Then for any i > N we have

2.1.5

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ϵ |α xi − αL| = |α| | xi − L| < |α|

= ϵ.

(2.1.33)

|α|

Thus lim

i→∞

α xi = αL

.

Q.E.D.

Proposition 2.1.6 Suppose {x

and {y } are convergent sequences in converges and

i }i∈I

{ xi + yi }

i∈I

i

i∈I

R

with

L = limi→∞ xi

and

M = limi→∞ yi .

Then the sequence

lim (xi + yi ) = L + M .

(2.1.34)

i→∞

Exercise 2.1.14 Prove the previous proposition.

Proposition 2.1.7 Suppose {x } and {y {x y } converges and i

i

i

i }i∈I

i∈I

are convergent sequences in

R

with

L = limi→∞ xi

and

M = limi→∞ yi .

Then the sequence

i∈I

lim xi yi = LM .

(2.1.35)

i→∞

Exercise 2.1.15 Prove the previous proposition.

Proposition 2.1.8 Suppose {x

i }i∈I

If M

≠ 0,

and {y

i }i∈I

are convergent sequences in R with L = lim

then the sequence {

i→∞

xi yi

}

xi , M = limi→∞ yi ,

and y

i

≠0

for all

i ∈ I.

converges and

i∈I

lim i→∞

xi

L =

yi

.

(2.1.36)

M

Proof Since M

≠0

and M

= limi→∞ yi ,

we may choose an integer N such that |M | | yi | >

(2.1.37) 2

whenever i > N . Let B be an upper bound for {|x P such that

i|

: i ∈ I } ∪ {| yi | : i ∈ I } .

M

2

Given any ϵ > 0, we may choose an integer

ϵ

| xi − L|

K, we have

2.1.6

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| xi M − yi L| ∣ xi L ∣ ∣ − ∣ = ∣ yi ∣ M | yi M | | xi M − xi yi + xi yi − yi L| = | yi M | | xi | |M − yi | + | yi | | xi − L| ≤ | yi M | B

M

2

ϵ

+B

4B

0 be given. Suppose L > 0 and note that − − − − − − − − | xi − L| = | √xi − √L || √xi + √L |

(2.1.44)

implies that − − − − | √xi − √L | =

| xi − L| (2.1.45)

− − − − | √xi + √L |

for any i ∈ I . Choose an integer N such that − − | xi − L| < √L ϵ

(2.1.46)

whenever i > N . Then, for any i > N ,

Hence lim

i→∞

− − − − √xi = √L

If L = 0, lim

i→∞

xi = 0,

− − √L ϵ

| xi − L|

− − − − | √xi − √L | =

− − < − − | √xi + √L |

= ϵ.

− − √L

(2.1.47)

.

so we may choose an integer N such that |x

i|

2

N . Then

− − | √xi | < ϵ

whenever i > N , so lim

i→∞

− − √xi = 0

.

(2.1.48)

Q.E.D.

Exercise 2.1.19 Evaluate − −−−− − √ 3 n2 + 1 lim

,

(2.1.49)

5n + 6

n→∞

carefully showing each step.

Exercise 2.1.20 Given real numbers r > 0 and α, show that (a)αr < r if 0 < α < 1 and (b)r < αr if α > 1 .

Proposition 2.1.10 If x ∈ R and |x| < 1, then n

lim x

= 0.

(2.1.50)

n→∞

Proof We first assume x ≥ 0 . Then the sequence converges. Let L = lim x . Then

n

is nonincreasing and bounded below by 0. Hence the sequence

∞

{x }

n=1

n

n→∞

n

L = lim x n→∞

n−1

= x lim x

= xL,

(2.1.51)

n→∞

2.1.8

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from which it follows that next exercise. Q.E.D.

since

L(1 − x) = 0.

1 − x > 0,

we must have

. The result for

L =0

follows from the

x N , let a = a and b = c; otherwise, let a = c and b =b . Let n = m and, for k = 2, 3, 4, … , let n be the smallest integer for which n > n and a ≤ x ≤ b Then {x } is a Cauchy sequence which is a subsequence of {x } . Thus { x } converges and Λ ≠ 0. Q.E.D. i−1

i

i−1

j

i

1

i

i

k

∞

nk

i−1

k

∞

i

k=1

k−1

k

nk

k

∞

nk

i=m

k=1

Exercise 2.1.26 Suppose A ⊂ R and B ⊂ R with a ≤ b for every a ∈ A and b ∈ B. Show that sup A ≤ inf B .

Proposition 2.1.12 Let Λ be the set of subsequential limits of a sequence {x

∞

i }i=m

.

Then

lim sup xi = sup Λ.

(2.1.54)

i→∞

Proof Let s = sup Λ and, for i ≥ m, u = sup {x : j ≥ i} . Now since x ≤ u for all j ≥ i, it follows that λ ∈ Λ and i ≥ m. Hence, from the previous exercise, s ≤ inf { u : i ≥ m} = lim sup x . i

j

j

i

i

λ ≤ ui

for every

i

i→∞

Now suppose s < lim sup x . Then there exists a real number t such that s < t < lim sup x . In particular, t t. For k = 2, 3, 4, … , let n be the smallest integer for which n > n and x > t. Then {x } is a subsequence of {x } which has a subsequential limit λ ≥ t . Since λ is also then a subsequential limit of {x } , we have λ ∈ Λ and λ ≥ t > s, contradicting s = sup Λ. Hence we must have lim sup x = sup Λ. Q.E.D. i→∞

i

i

i→∞

1

1

n1

k

∞

k

k−1

nk

nk

i

∞

i

k=1

i=m

∞

i

i→∞

i=m

i

Exercise 2.1.27 Let Λ be the set of subsequential limits of a sequence {x

∞

i }i=m

.

Show that

lim inf xi = inf Λ.

(2.1.55)

i→∞

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2.1.10

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2.2: Infinite Series Definition ∞

Given a sequence {a

i }i=m

,

let {s

∞

n }n=m

be the sequence defined by n

sn = ∑ ai .

(2.2.1)

i=m ∞

We call the sequence {s

∞

an infinite series. If {s

n }n=m

converges, we call

n }n=m

s = lim sn

(2.2.2)

n→∞

the sum of the series. For any integer n ≥ m, we call s a partial sum of the series. n

We will use the notation ∞

∑ ai

(2.2.3)

i=1

to denote either {s ∑ a diverges.

the infinite series, or s , the sum of the infinite series. Of course, if {s

∞

n }n=m ,

diverges, then we say

∞

n }n=m

∞

i=m

i

Exercise 2.2.1 Suppose ∑

∞ i=m

ai

converges and β ∈ R. Show that ∑

∞ i=m

β ai

also converges and

∞

∞

∑ β ai = β ∑ ai . i=m

(2.2.4)

i=m

Exercise 2.2.2 Suppose both ∑

∞ i=m

ai

and ∑

∞ i=m

bi

converge. Show that ∑

∞ i=m

∞

(ai + bi )

∞

converges and

∞

∑ (ai + bi ) = ∑ ai + ∑ bi . i=m

i=m

(2.2.5)

i=m

Exercise 2.2.3 Given an infinite series ∑

∞ i=m

ai

and an integer k ≥ m, show that ∑

∞

ai

i=m

converges if and only if ∑

∞ i=k

ai

converges.

Proposition 2.2.1 Suppose ∑

∞ i=m

ai

converges. Then lim

n→∞

an = 0

.

Proof Let

n

sn = ∑

i=m

ai

and

Since

s = limn→∞ sn .

an = sn − sn−1 ,

limn→∞ an = limn→∞ (sn − sn−1 ) = limn→∞ sn − limn→∞ sn−1 = s − s = 0

we

have

. Q.E.D.

Exercise 2.2.4 ∞

Let s = ∑

i=0

n

(−1 ) .

Note that ∞

∞ n

s = ∑(−1 ) n=0

n

= 1 − ∑(−1 )

= 1 − s,

(2.2.6)

n=0

2.2.1

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from which it follows that s =

1 2

Is this correct?

.

Exercise 2.2.5 Show that for any real number x ≠ 1 , n

n+1

i

sn = ∑ x

1 −x =

.

(Hint: Note that x

n+1

(2.2.7)

1 −x

i=0

= sn+1 − sn = 1 + x sn − sn . )

Theorem 2.2.2 For any real number x with |x| < 1, ∞ n

∑x

1 =

.

(2.2.8)

1 −x

n=0

Proof If s

n

n

=∑

i=0

i

x ,

then, by the previous exercise, n+1

1 −x sn =

.

(2.2.9)

1 −x

Hence ∞

n+1

n

∑x

1 −x = lim sn = lim n→∞

n→∞

n=0

1 =

1 −x

.

(2.2.10)

1 −x

2.2.1 Comparison Tests The following two propositions are together referred to as the comparison test.

Proposition 2.2.3 ∞

∞

Suppose ∑ a and ∑ b are infinite series for which there exists an integer N such that 0 ≤ a If ∑ b converges, then ∑ a converges. i=m

∞

i=k

i

i

i=k

≤ bi

i

whenever i ≥ N .

∞

i

i=m

i

Proof By Exercise 2.2 .3 We need only show that ∑ nth partial sum of ∑ b . Now

∞ i=N

ai

converges. Let s be the nth partial sum of ∑

∞

n

i=N

ai

and let t be the n

∞

i

i=N

sn+1 − sn = an+1 ≥ 0 ∞

for every n ≥ N , so {s

n }n=N

(2.2.11)

is a nondecreasing sequence. Moreover, ∞

sn ≤ tn ≤ ∑ bi < +∞

(2.2.12)

i=N

for every n ≥ N . Thus {s

∞

n }n=N

is a nondecreasing, bounded sequence, and so converges.

Proposition 2.2.4 Suppose ∑ a and ∑ If ∑ a diverges then ∑ ∞

i=m

∞

i=k

i

∞

i

bi i=k ∞ i=m

are infinite series for which there exists an integer N such that 0 ≤ a b diverges.

i

≤ bi

whenever i ≥ N .

i

Proof

2.2.2

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By Exercise 2.2 .3 we need only show that ∑ b diverges. Let s be the nth partial sum of ∑ a and let t be the nth partial sum of ∑ b . Now { s } , N is a nondecreasing sequence which diverges, and so we must have lim s = +∞. Thus given any real number M there exists an integer L such that ∞

∞

i=N

∞

i=N

n→∞

i

n

i=N

i

n

∞

i

n

n=N

n

M < sn ≤ tn

whenever n > L. Hence lim

n→∞

tn = +∞

and ∑

∞ i=m

(2.2.13)

diverges. (\quad\) Q.E.D.

bi

Example 2.2.1 Consider the infinite series ∞

1

∑ n=0

1

1

= 1 +1 +

+

n!

1 +

2

3!

+⋯ .

(2.2.14)

4!

Now for n = 1, 2, 3, … , we have 1

1

0

0

such that

∞

i

i=k

i

Proof By the comparison test, ∑

∞ i=m

M bi

diverges. Hence, by the previous exercise, ∑

∞ i=m

2.2.4

bi

also diverges.

https://math.libretexts.org/@go/page/22644

We call the results of the next two exercises, which are direct consequences of the last two propositions, the limit comparison test.

Exercise 2.2.7 Suppose ∑ and

∞ i=m

ai

and ∑

∞ i=m

bi

are infinite series for which a

i

ai

lim

∞

>0

i

for all i ≥ m. Show that if ∑

∞ i=m

bi

converges

< +∞,

(2.2.21)

converges.

ai

i=m

and b

bi

i→∞

then ∑

≥0

Exercise 2.2.8 Suppose ∑ and

∞ i=m

ai

and ∑

∞ i=m

bi

are infinite series for which a

i

ai

lim

∞ i=m

bi

and b

i

>0

for all i ≥ m. Show that if

∞

∑

i=m

< +∞,

ai

diverges

(2.2.22)

bi

i→∞

then ∑

≥0

diverges.

Exercise 2.2.9 Show that ∞

1

∑ n=1

(2.2.23)

n

n2

converges.

Exercise 2.2.10 Show that ∞

n

x

∑ n=0

(2.2.24) n!

converges for any real number x ≥ 0 .

Exercise 2.2.11 Let S be the set of all finite sums of numbers in the set {a

1,

a2 , a3 , …} ,

where a

i

≥0

for i = 1, 2, 3, … That is +

S = { ∑ ai : J ⊂ {1, 2, 3, … , n} for some n ∈ Z

}.

(2.2.25)

i∈J

Show that ∑

∞ i=1

ai

converges if and only if sup S < ∞, in which case ∞

∑ ai = sup S.

(2.2.26)

i=1

One consequence of the preceding exercise is that the sum of a sequence of nonnegative numbers depends only on the numbers begin added, and not on the order in which they are added. That is, if φ : Z → Z is one-to-one and onto, ∑ a converges if and only if ∑ a converges, and, in that case, +

+

∞

i=1

i

∞

i=1

φ(i)

2.2.5

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∞

∞

∑ ai = ∑ aφ(i) . i=1

(2.2.27)

i=1

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2.2.6

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CHAPTER OVERVIEW 3: Cardinality 3.1: Binary Representations 3.2: Countable and Uncountable Sets 3.3: Power Sets Thumbnail: Bijective function from N to the set E of even numbers. Although E is a proper subset of N, both sets have the same cardinality. (CC BY-SA 3.0; Jochen Burghardt via Wikipedia) This page titled 3: Cardinality is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1

3.1: Binary Representations ∞

Suppose {a } is a sequence such that, for each n = 1, 2, 3, … , either a integer n > N such that a = 0. Then n

n

n=1

or a

=0

=1

n

and, for any integer N , there exists an

n

an

0 ≤

1 ≤

n

(3.1.1)

n

2

2

for n = 1, 2, 3, … , so the infinite series ∞

an

∑

(3.1.2)

n

2

n=1

converges to some real number x by the comparison test. Moreover, ∞

1

0 ≤x 1 and s

n−1

1 sn = sn−1 +

If x < s

n−1

+

1 n

2

,

then a

n

=0

n

2

1 2

1 2

≤ x < sn−1 +

1 ≤ x < sn−1 +

(3.1.8)

≤x 0, choose an integer N such that

1 n

2

< ϵ.

Then, for any n > N , it follows from the lemma that 1

| sn − x|

N . Then

0 ≤ x < 1.

∑ n=N +1

implying that a

N

= 1,

= 1,

(3.1.14)

Now suppose there exists an integer

∞

x = sN +

n

2

∞

1 n

= sN −1 +

2

1

∑

and thus contradicting the assumption that a

N

= sN −1 +

2

=0

such that

aN = 0

but

an = 1

for

1

n

n=N +1

N

,

(3.1.15)

N

2

.

Q.E.D.

Combining the previous lemma with the previous proposition yields the following result.

Proposition 3.1.4 With the notation as above, x =. a

1 a2 a3 a4

…

.

The next theorem now follows from Exercise 3.1.1 and the previous proposition.

Theorem 3.1.5 Every real number 0 ≤ x < 1 has a unique binary representation.

3.1.2

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3.1.3

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3.2: Countable and Uncountable Sets Definition A function φ : A → B is said to be a one-to-one correspondence if φ is both one-to-one and onto.

Definition We say sets A and B have the same cardinality if there exists a one-to-one correspondence φ : A → B . We denote the fact A and B have the same cardinality by writing |A| = |B|.

Exercise 3.2.1 Define a relation on sets by setting A ∼ B if and only if |A| = |B|. Show that this relation is an equivalence relation.

Definition Let A be a set. If, for n ∈ Z , A has the cardinality of the set {1, 2, 3, … , n},we say A is finite and write |A| = n. If A has the cardinality of Z , we say A is countable and write |A| = ℵ . +

+

0

Example 3.2.1 If we define φ : Z

+

→ Z

by n−1

φ(n) = {

,

2

−

n 2

if n is odd, (3.2.1)

,

if n is even,

then φ is a one-to-one correspondence. Thus |Z| = ℵ . 0

Exercise 3.2.2 Let A be the set of even integers. Show that |A| = ℵ . 0

Exercise 3.2.3 Verify each of the following: a. If A is a nonempty subset of Z

+

,

then A is either finite or countable.

b. If A is a nonempty subset of a countable set B, then A is either finite or countable.

Proposition 3.2.1 Suppose A and B are countable sets. Then the set C

is countable.

= A∪B

Proof Suppose A and B are disjoint, that is, Define τ : Z → C by

A ∩ B = ∅.

Let

+

φ : Z

→ A

and

+

ψ : Z

→ B

be one-to-one correspondences.

+

⎧φ( τ (n) = ⎨ ⎩

ψ(

n+1 2 n 2

),

),

if n is odd, (3.2.2) if n is even.

Then τ is a one-to-one correspondence, showing that C is countable.

3.2.1

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If A and B are not disjoint, then τ is onto but not one-to-one. However, in that case subset of Z , and so is countable. Q.E.D.

C

has the cardinality of an infinite

+

Definition A nonempty set which is not finite is said to be infinite. An infinite set which is not countable is said to be uncountable.

Exercise 3.2.4 Suppose A is uncountable and B ⊂ A is countable. Show that A∖B is uncountable.

Proposition 3.2.2 Suppose A and B are countable. Then C

= A×B

is countable.

Proof Let φ : Z letting

+

→ A

and ψ : Z

+

→ B

be one-to-one correspondences. Let a

i

= φ(i)

and b

i

= ψ(i).

Define τ

+

: Z

→ C

by

τ (1) = (a1 , b1 ) ,

(3.2.3)

τ (2) = (a1 , b2 ) ,

(3.2.4)

τ (3) = (a2 , b1 ) ,

(3.2.5)

τ (4) = (a1 , b3 ) ,

(3.2.6)

τ (5) = (a2 , b2 ) ,

(3.2.7)

τ (6) = (a3 , b1 ) ,

(3.2.8)

τ (7) = (a1 , b4 ) ,

(3.2.9)

⋮ =⋮

(3.2.10)

That is, form the infinite matrix with (a , b ) in the ith row and j th column, and then count the entries by reading down the diagonals from right to left. Then τ is a one-to-one correspondence and C is countable. i

j

Proposition 3.2.3 Q

is countable.

Proof By the previous proposition, Z × Z is countable. Let A = {(p, q) : p, q ∈ Z, q > 0, p and q relatively prime . }

(3.2.11)

Then A is infinite and A ⊂ Z × Z, so A is countable. But clearly |Q| = |A|, so Q is countable.

Q.E.D.

Proposition 3.2.4 Suppose for each i ∈ Z

+

, Ai

is countable. Then ∞

B = ⋃ Ai

(3.2.12)

i=1

is countable.

3.2.2

https://math.libretexts.org/@go/page/22651

Proof Suppose the sets A , i ∈ Z , are pairwise disjoint, that is, A φ : Z → A be a one-to-one correspondence. Then ψ : Z × Z +

i

i

+

+

i

+

i

for all defined by

∩ Aj = ∅ → B

+

i, j ∈ Z

For each

.

ψ(i, j) = φi (j)

is a one-to-one correspondence, and so |B| = ∣∣Z

+

+

×Z

∣ ∣ = ℵ0

+

i ∈ Z

,

let

(3.2.13)

.

If the sets A , i ∈ Z , are not disjoint, then ψ is onto but not one-to-one. But then there exists a subset P of Z × Z such that ψ : P → B is a one-to-one correspondence. Since P is an infinite subset of a countable set, P is countable and so |B| = ℵ . Q.E.D. +

+

+

i

0

If in the previous proposition we allow that, for each i ∈ Z or countable.

+

, Ai

is either finite or countable, then B = ⋃

∞ i=1

Ai

will be either finite

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3.2.3

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3.3: Power Sets Definition Given a set A, we call the set of all subsets of A the power set of A, which we denote P(A) .

Example 3.3.1 If A = {1, 2, 3}, then P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

(3.3.1)

Proposition 3.3.1 If A is finite with |A| = n, then |P(A)| = 2 . n

Proof Let B = {(b1 , b2 , … , bn ) : bi = 0 or bi = 1, i = 1, 2, … , n}

and let a

1,

(3.3.2)

be the elements of A. Define φ : B → P(A) by letting

a2 , … , an

φ (b1 , b2 , … , bn ) = { ai : bi = 1, i = 1, 2, … , n} .

Then φ is a one-to-one correspondence. The result now follows from the next exercise.

(3.3.3)

Q.E.D.

Exercise 3.3.1 With B as in the previous proposition, show that |B| = 2 . n

In analogy with the case when A is finite, we let 2

|A|

= |P(A)|

for any nonempty set A.

Definition Suppose A and B are sets for which there exists a one-to-one function correspondence ψ : A → B. Then we write |A| < |B|.

¯ φ : A → B

but there does not exist a one-to-one

Theorem 3.3.2 If A is a nonempty set, then |A| < |P(A)| . Proof Define φ : A → P (Z

+

)

by ∞

φ ({ ai }

i=1

Then φ is a one-to-one correspondence. Now let B be the set of all sequences {a Let C = A∖B ,

∞

i }i=1

+

) = {i : i ∈ Z

, ai = 1} .

(3.3.4)

Q.E.D. in A such that for every integer N there exists an integer n > N such that ∞

D0 = { { ai }

i=1

: ai = 1, i = 1, 2, 3, …} ,

an = 0.

(3.3.5)

and

3.3.1

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∞

Dj = { { ai }

: aj = 0, ak = 1 for k > j}

i=1

for j = 1, 2, 3, … Then |D

0|

=1

and |D

j|

j−1

=2

(3.3.6)

for j = 1, 2, 3, … Moreover, ∞

C = ⋃ Dj ,

(3.3.7)

j=0

so C is countable. Since A = B ∪ C , and A is uncountable, it follows that B is uncountable. Now if we let I = {x : x ∈ R, 0 ≤ x < 1},

(3.3.8)

we have seen that the function φ : B → I defined by ∞

φ ({ ai }

i=1

) = a1 a2 a3 a4 …

(3.3.9)

is a one-to-one correspondence. It follows that I is uncountable. As a consequence, we have the following result.

Theorem 3.3.4 R

is uncountable.

Exercise 3.3.2 Let I

Show that

= {x : x ∈ R, 0 ≤ x < 1}.

a. |I | = |{x : x ∈ R, 0 ≤ x ≤ 1}| b. |I | = |{x : x ∈ R, 0 < x < 1}| c. |I | = |{x : x ∈ R, 0 < x < 2}| d. |I | = |{x : x ∈ R, −1 < x < 1}|

Exercise 3.3.3 Let I

= {x : x ∈ R, 0 ≤ x < 1}

and suppose a and b are real numbers with a < b. Show that |I | = |{x : x ∈ R, a ≤ x < b}|.

(3.3.10)

Exercise 3.3.4 Does there exist a set A ⊂ R for which about Cantor's continum hypothesis.)

ℵ0

ℵ0 < |A| < 2

?

(Before working too long on this problem, you may wish to read

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CHAPTER OVERVIEW 4: Topology of the Real Line 4.1: Intervals 4.2: Open Sets 4.3: Closed Sets 4.4: Compact Sets Thumbnail: The blue circle represents the set of points (x, y) satisfying x + y = r . The red disk represents the set of points (x, y) satisfying x + y < r . The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set. (Public Domain; Richard Giuly via Wikipedia) 2

2

2

2

2

2

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1

4.1: Intervals Definition: intervals Given any two extended real numbers a < b, we call the set (a, b) = {x : x ∈ R, a < x < b}

(4.1.1)

an open interval. Given any two finite real numbers a ≤ b, we call the sets [a, b] = {x : x ∈ R, a ≤ x ≤ b},

(4.1.2)

(−∞, b] = {x : x ∈ R, x ≤ b},

(4.1.3)

[a, +∞) = {x : x ∈ R, x ≥ a}

(4.1.4)

and

closed intervals. We call any set which is an open interval, a closed interval, or is given by, for some finite real numbers a < b , (a, b] = {x : x ∈ R, a < x ≤ b}

(4.1.5)

[a, b) = {x : x ∈ R, a ≤ x < b},

(4.1.6)

(a, b) = {x : x = λa + (1 − λ)b, 0 < λ < 1}.

(4.1.7)

or

an interval.

Proposition 4.1.1 If a, b ∈ R with a < b, then

Proof Suppose x = λa + (1 − λ)b for some 0 < λ < 1. Then b − x = λb − λa = λ(b − a) > 0,

(4.1.8)

x − a = (λ − 1)a + (1 − λ)b = (1 − λ)(b − a) > 0,

(4.1.9)

so x < b. Similarly,

so a < x. Hence x ∈ (a, b) . Now suppose x ∈ (a, b). Then b −x x =(

x −a ) a+(

b −a

b −x )b =(

b −a

b −x ) a + (1 −

b −a

)b

(4.1.10)

b −a

and b −x 0

0 such that (x − ϵ, x + ϵ) ⊂ U .

(4.2.1)

Proposition 4.2.1 Every open interval I is an open set. Proof Suppose I = (a, b), where a < b are extended real numbers. Given x ∈ I , let Suppose y ∈ (x − ϵ, x + ϵ). If b = +∞, then b > y; otherwise, we have

ϵ

be the smaller of

x −a

and

b − y > b − (x + ϵ) = (b − x) − ϵ ≥ (b − x) − (b − x) = 0,

b − x.

(4.2.2)

so b > y. If a = −∞, then a < y; otherwise, y − a > (x − ϵ) − a = (x − a) − ϵ ≥ (x − a) − (x − a) = 0,

so a < y. Thus y ∈ I and I is an open set.

(4.2.3)

Q.E.D.

Note that R is an open set (it is, in fact, the open interval (−∞, +∞)), as is ∅ (it satisfies the definition trivially).

Proposition 4.2.2 Suppose A is a set and, for each α ∈ A, U is an open set. Then α

⋃ Uα

(4.2.4)

α∈A

is an open set. Proof Let x ∈ ∪ Thus

α∈A

Uα .

Then x ∈ U for some α ∈ A. Since U is open, there exists an α

α

ϵ>0

such that

(x − ϵ, x + ϵ) ⊂ Uα .

(x − ϵ, x + ϵ) ⊂ Uα ⊂ ⋃ Uα .

(4.2.5)

α∈A

Hence ⋃

α∈A

Uα

is open.

Q.E.D.

Proposition 4.2.3 Suppose U

1,

U2 , … , Un

is a finite collection of open sets. Then n

⋂ Ui

(4.2.6)

i=1

is open. Proof n

Let x ∈ ⋂ U . Then x ∈ U for every i = 1, 2, … , n. For each i, choose ϵ be the smallest of ϵ , ϵ , … , ϵ . Then ϵ > 0 and i=1

i

i

1

2

i

>0

such that (x − ϵ

i,

x + ϵi ) ⊂ Ui .

Let ϵ

n

(x − ϵ, x + ϵ) ⊂ (x − ϵi , x + ϵi ) ⊂ Ui

4.2.1

(4.2.7)

https://math.libretexts.org/@go/page/22658

for every i = 1, 2, … , n. Thus n

(x − ϵ, x + ϵ) ⊂ ⋂ Ui .

(4.2.8)

i=1

Hence ⋂

n i=1

Ui

is an open set.

Q.E.D.

Definition Let A ⊂ R. We say x ∈ A is an interior point of A if there exists an ϵ > 0 such that (x − ϵ, x + ϵ) ⊂ A. We call the set of all interior points of A the interior of A, denoted A . ∘

Exercise 4.2.1 Show that if A ⊂ R, then A is open. ∘

Exercise 4.2.2 Show that A is open if and only if A = A . ∘

Exercise 4.2.3 Let U

⊂R

be a nonempty open set. Show that sup U

∉ U

and inf U

∉ U

.

This page titled 4.2: Open Sets is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

4.2.2

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4.3: Closed Sets Definition: limit points We call a point x ∈ R a limit point of a set A ⊂ R if for every ϵ > 0 there exists a ∈ A, a ≠ x, such that a ∈ (x − ϵ, x + ϵ) .

Definition: isolated point Suppose A ⊂ R. We call a point a ∈ A an isolated point of A if there exists an ϵ > 0 such that A ∩ (a − ϵ, a + ϵ) = {a}.

(4.3.1)

Exercise 4.3.1 Identify the limit points and isolated points of the following sets: a. [−1, 1], b. (−1, 1), c. {

1 n

+

: n ∈ Z

}

,

d. Z, e. Q.

Exercise 4.3.2 Suppose x is a limit point of the set A. Show that for every ϵ > 0, the set (x − ϵ, x + ϵ) ∩ A is infinite. We let A denote the set of limit points of a set A. ′

Definition: closures ¯ Given a set A ⊂ R, we call the set A = A∪A the closure of A . ′

Definition: Closed sets We call a set C

⊂R

closed if C

¯ =C

.

Proposition 4.3.1 ¯ If A ⊂ R, then A is closed.

Proof ¯ ¯ Suppose x is a limit point of A . We we will show that x is a limit point of A, and hence x ∈ A. Now for any ϵ > 0, there ¯ exists a ∈ A, a ≠ x, such that ϵ a ∈ (x −

ϵ ,x+

2

).

(4.3.2)

2

If a ∈ A, let b = a. If a ∉ A, then a is a limit point of A, so there exists b ∈ A, b ≠ a and b ≠ x, such that ϵ b ∈ (a −

ϵ ,a+

2

).

(4.3.3)

2

In either case

4.3.1

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ϵ |x − b| ≤ |x − a| + |a − b| < 2 ¯ Hence x ∈ A , and so A is closed. ′

ϵ +

= ϵ.

(4.3.4)

2

Q.E.D.

Proposition 4.3.2 A set C

⊂R

is closed if and only if for every convergent sequence {a

k }k∈K

with a

k

∈ C

for all k ∈ K ,

lim ak ∈ C .

(4.3.5)

k→∞

Proof Suppose C is closed and for some integer k, then a ≠ x and

{ ak }

is a convergent sequence with a ∈ C for all k ∈ K. Let x = lim Otherwise, for every ϵ > 0, there exists an integer N such that |a k

k∈K

x ∈ C.

k→∞

N

ak .

If x = a . Hence k

− x| < ϵ

N

aN ∈ (x − ϵ, x + ϵ).

(4.3.6)

Thus x is a limit point of C , and so x ∈ C since C is closed. Now suppose that for every convergent sequence point of C . For k = 1, 2, 3, … , choose a

k

∈ C

with a

{ ak }

k

k∈K

such that a

k

∈ C

∈ (x −

1 k

for all

,x+

1 k

k ∈ K, limk→∞ ak ∈ C .

).

Let

x

be a limit

Then clearly

x = lim ak ,

(4.3.7)

k→∞

so x ∈ C . Thus C is closed.

Q.E.D.

Exercise 4.3.3 Show that every closed interval I is a closed set.

Proposition 4.3.3 Suppose A is a set and, for each α ∈ A, C is a closed set. Then α

⋂ Cα

(4.3.8)

α∈A

is a closed set. Proof Suppose

is a limit point of ⋂ C . Then for any ϵ > 0, there exists y ∈ ⋂ But then for any α ∈ A, y ∈ C , so x is a limit point of C . Since for every α ∈ A. Thus x ∈ ⋂ C and ⋂ C is closed. Q.E.D. x

α∈A

α

α∈A

y ∈ (x − ϵ, x + ϵ). x ∈ Cα

α

α∈A

α

α

such that y ≠ x and is closed, it follows that

Cα

Cα

α

α∈A

Proposition 4.3.4 Suppose C

1,

C2 , … , Cn

is a finite collection of closed sets. Then n

⋃ Ci

(4.3.9)

i=1

is closed. Proof

4.3.2

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Suppose {a } is a convergent sequence with a ∈ ⋃ infinite set, there must an integer m and a subsequence

n

k

k

k∈K

subsequence of {a

k }k∈K

converges to L, {a

nj

∞

}

j=1

i=1

Ci

{ anj

for every k ∈ K. Let L = lim a . Since K is an } such that a ∈ C for j = 1, 2, … . Since every k→∞

k

∞

nj

j=1

m

must converge to L. Since C is closed, m

n

L = lim an j→∞

n

Thus ⋃

i=1

Ci

is closed.

j

∈ Cm ⊂ ⋃ Ci .

(4.3.10)

i=1

Q.E.D.

Note that both R and ∅ satisfy the definition of a closed set.

Proposition 4.3.5 A set C

⊂R

is closed if and only if R∖C is open.

Proof Assume C is closed and let U = R∖C . If C = R, then U = ∅, which is open; if C = ∅, then U = R, which is open. So we may assume both C and U are nonempty. Let x ∈ U . Then x is not a limit point of C , so there exists an ϵ > 0 such that (x − ϵ, x + ϵ) ∩ C = ∅.

(4.3.11)

(x − ϵ, x + ϵ) ⊂ U ,

(4.3.12)

Thus

so U is open. Now suppose U = R∖C is open. If U = R, then C = ∅, which is closed; if U = ∅, then C may assume both U and C are nonempty. Let x be a limit point of C . Then, for every ϵ > 0 ,

= R,

which is closed. So we

(x − ϵ, x + ϵ) ∩ C ≠ ∅.

(4.3.13)

(x − ϵ, x + ϵ) ⊂ U .

(4.3.14)

Hence there does not exist ϵ > 0 such that

Thus x ∉ U , so x ∈ C and C is closed.

Q.E.D.

Exercise 4.3.4 For n = 1, 2, 3, … , let I

n

= (−

1 n

,

n+1 n

).

Is ∞

⋂ In

(4.3.15)

n=1

open or closed?

Exercise 4.3.5 For n = 3, 4, 5, … , let I

n

=[

1 n

,

n−1 n

].

Is ∞

⋃ In

(4.3.16)

n=3

open or closed?

4.3.3

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Exercise 4.3.6 Suppose, for

the intervals show that

n = 1, 2, 3, … , +

b = inf { bn : n ∈ Z

},

In = [ an , bn ]

are such that

In+1 ⊂ In .

If

+

a = sup { an : n ∈ Z

}

and

∞

⋂ In = [a, b].

(4.3.17)

n=1

Exercise 4.3.7 Find a sequence I

n,

n = 1, 2, 3, … ,

of closed intervals such that I

n+1

⊂ In

for n = 1, 2, 3, … and

∞

⋂ In = ∅.

(4.3.18)

n=1

Exercise 4.3.8 Find a sequence I

n,

n = 1, 2, 3, … ,

of bounded, open intervals such that I

n+1

⊂ In

for n = 1, 2, 3, … and

∞

⋂ In = ∅.

(4.3.19)

n=1

Exercise 4.3.9 Suppose A

i

⊂ R, i = 1, 2, … , n,

and let B = ⋃

n i=1

Show that

Ai .

n ¯ ¯¯ ¯

¯¯¯¯ ¯

B = ⋃ Ai .

(4.3.20)

i=1

Exercise 4.3.10 Suppose A

+

i

⊂ R, i ∈ Z

,

and let ∞

B = ⋃ Ai .

(4.3.21)

i=1

Show that ∞ ¯¯¯¯ ¯

¯ ¯¯ ¯

⋃ Ai ⊂ B.

(4.3.22)

i=1

Find an example for which ∞ ¯ ¯¯ ¯

¯¯¯¯ ¯

B ≠ ⋃ Ai .

(4.3.23)

i=1

Exercise 4.3.11 Suppose U

⊂R

is a nonempty open set. For each x ∈ U , let Jx = ⋃(x − ϵ, x + δ),

(4.3.24)

where the union is taken over all ϵ > 0 and δ > 0 such that (x − ϵ, x + δ) ⊂ U . a. Show that for every x, y ∈ U , either J

x

∩ Jy = ∅

or J

x

= Jy

.

4.3.4

https://math.libretexts.org/@go/page/22659

b. Show that U = ⋃ Jx ,

(4.3.25)

x∈B

where B ⊂ U is either finite or countable. This page titled 4.3: Closed Sets is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

4.3.5

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4.4: Compact Sets Definition Suppose T

⊂ R.

If A is a set, U is an open set for every α ∈ A, and α

T ⊂ ⋃ Uα ,

(4.4.1)

α∈A

then we call {U

α

: α ∈ A}

an open cover of T .

Example For n = 3, 4, 5, … , let 1 Un = (

Then {U

n

: n = 3, 4, 5, …}

n−1 ,

).

n

(4.4.2)

n

is an open cover of the open interval (0, 1).

Definition Suppose {U

α

: α ∈ A}

is an open cover of T

⊂ R.

If B ⊂ A and T ⊂ ⋃ Uβ ,

(4.4.3)

β∈B

then we call

{ Uβ : β ∈ B}

{ Uα : α ∈ A}

.

a subcover of

{ Uα : α ∈ A} .

If

B

is finite, we call

{ Uβ : β ∈ B}

a finite subcover of

Exercise Show that the open cover of (0, 1) given in the previous example does not have a finite subcover.

Definition We say a set K ⊂ R is compact if every open cover of K has a finite sub cover.

Example As a consequence of the previous exercise, the open interval (0, 1) is not compact.

Exercise Show that every finite subset of R is compact.

Exercise Suppose n ∈ Z

+

and K

1,

K2 , … , Kn

n

are compact sets. Show that ⋃

i=1

Ki

is compact.

proposition If I is a closed, bounded interval, then I is compact. Proof Let

a ≤b

be finite real numbers and I = [a, b]. Suppose {U : α ∈ A} is an open cover of with the properties that B is a finite subset of A and a ∈ ⋃ U . Let α

{ Uβ : β ∈ B}

β∈B

4.4.1

I.

Let

O

be the set of sets

β

https://math.libretexts.org/@go/page/22660

(s − ϵ, s + ϵ) ⊂ Uα .

Moreover, there exists a {U

β

: β ∈ B} ∈ O

(4.4.4)

for which ϵ [a, s −

] ⊂ ⋃ Uβ . 2

(4.4.5)

β∈B

But then { Uβ : β ∈ B} ∪ { Uα } ∈ O

(4.4.6)

and ϵ [a, s + 2

] ⊂ ( ⋃ Uβ ) ∪ Uα ,

(4.4.7)

β∈B

contradicting the definition of s. Hence we must have s = b. Now choose U such that b ∈ U

α.

α

Then, for some ϵ > 0 ,

(b − ϵ, b + ϵ) ⊂ Uα .

Moreover, there exists {U

β

: β ∈ B} ∈ O

(4.4.8)

such that ϵ [a, b − 2

] ⊂ ⋃ Uβ .

(4.4.9)

β∈B

Then { Uβ : β ∈ B} ∪ { Uα } ∈ O

is a finite subcover of I . Thus I is compact.

(4.4.10)

Q.E.D.

proposition If K is a closed, bounded subset of R, then K is compact. Proof Since K is bounded, there exist finite real numbers a and b such that K. Let V = R∖K. Then

K ⊂ [a, b].

Let {U

α

: α ∈ A}

be an open cover of

{ Uα : α ∈ A} ∪ {V }

(4.4.11)

K ⊂ [a, b]∖V ⊂ ⋃ Uβ .

(4.4.12)

in the latter case, we have

β∈B

In either case, we have found a finite subcover of {U

α

: α ∈ A}

.

Q.E.D.

Exercise Show that if K is compact and C

⊂K

is closed, then C is compact.

proposition If K ⊂ R is compact, then K is closed. Proof Suppose x is a limit point of K and x ∉ K. For n = 1, 2, 3, … , let 1 Un = (−∞, x −

1 ) ∪ (x +

n

4.4.2

, +∞) .

(4.4.13)

n

https://math.libretexts.org/@go/page/22660

Then ∞

⋃ Un = (−∞, x) ∪ (x, +∞) ⊃ K.

(4.4.14)

n=1

However, for any N

+

∈ Z

,

there exists a ∈ K with 1

1

a ∈ (x −

,x+ N

),

(4.4.15)

N

and hence N

1

a ∉ ⋃ Un = (−∞, x − n=1

Thus the open cover {U Q.E.D.

+

: n ∈ Z

n

}

1 ) ∪ (x +

N

, +∞) .

(4.4.16)

N

does not have a finite subcover, contradicting the assumption that K is compact.

Exercise Suppose that for each α in some set A, K is compact. Show that ⋂ α

Kα

α∈A

is compact.

proposition If K ⊂ R is compact, then K is bounded. Proof Suppose K is not bounded. For n = 1, 2, 3, … , let U

n

= (−n, n).

Then

∞

⋃ Un = (−∞, ∞) ⊃ K.

(4.4.17)

n=1

But, for any integer N , there exists a ∈ K such that |a| > N , from which it follows that N

a ∉ ⋃ Un = (−N , N ).

(4.4.18)

n=1

Thus the open cover {U Q.E.D.

+

: n ∈ Z

n

}

does not have a finite subcover, contradicting the assumption that K is compact.

Taken together, the previous three propositions yield the following fundamental result:

Theorem A set K ⊂ R is compact if and only if K is closed and bounded.

proposition If K ⊂ R is compact and {x {x } with

n }n∈I

is a sequence with x

n

∈ K

for every n ∈ I , then {x

n }n∈I

has a convergent subsequence

∞

nk

k=1

lim xnk ∈ K.

(4.4.19)

k→∞

Proof Since

K

is bounded, ∈ K.

{ xn }

n∈I

has a convergent subsequence

∞

{ xn } k

k=1

. Since

K

is closed, we must have

limk→∞ xnk

4.4.3

https://math.libretexts.org/@go/page/22660

proposition Suppose K ⊂ R is such that whenever {x } is a sequence with x {x } with lim x ∈ K. Then K is compact. n

n

n∈I

∈ K

for every n ∈ I , then {x

n }n∈I

has a subsequence

∞

nk

k→∞

k=1

nk

Proof ∞

Suppose K is unbounded. Then we may construct a sequence {x } such that x ∈ K and |x | > n for n = 1, 2, 3, … Hence the only possible subsequential limits of {x } would be −∞ and +∞, contradicting our assumptions. Thus K must be bounded. n

n=1

n

n

∞

n

n=1

Now suppose {x } is a convergent sequence with x ∈ K for all n ∈ I . If L = lim x , then subsequential limit of {x } . Hence, by the assumptions of the proposition, L ∈ K. Hence K is closed. n

n

n∈I

n

n→∞

n

L

is the only

n∈I

Since K is both closed and bounded, it is compact.

Q.E.D.

Exercise Show that a set K ⊂ R is compact if and only if every infinite subset of K has a limit point in K .

Exercise Show that if K is compact, then sup K ∈ K and inf K ∈ K .

Theorem Given a set K ⊂ R, the following are equivalent: 1. Every open cover of K has a finite subcover. 2. Every sequence in K has a subsequential limit in K . 3. Every infinite subset of K has a limit point in K .

Exercise Suppose K

1,

K2 , K3 , …

are nonempty compact sets with Kn+1 ⊂ Kn

(4.4.20)

for n = 1, 2, 3, … Show that ∞

⋂ Kn

(4.4.21)

n=1

is nonempty.

Exercise We say a collection of sets {D

α

: α ∈ A}

has the finite intersection property if for every finite set B ⊂ A , ⋂ Dα ≠ ∅.

(4.4.22)

α∈B

Show that a set K ⊂ R is compact if and only for any collection { Eα : α ∈ A, Eα = Cα ∩ K where Cα ⊂ R is closed }

(4.4.23)

which has the finite intersection property we have ⋂ Eα ≠ ∅.

(4.4.24)

α∈A

4.4.4

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4.4.5

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CHAPTER OVERVIEW 5: Limits and Continuity 5.1: Limits 5.2: Monotonic Functions 5.3: Limits to Infinity and Infinite Limits 5.4: Continuous Functions Thumbnail: The function f (x) = 1/(x − a) has infinite limits at a. (CC BY; OpenStax) n

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1

5.1: Limits Let

and let x be a limit point of A. In the following, we will let S(A, x) denote the set of all convergent sequences {x } such that x ∈ A for all n ∈ I , x ≠ x for all n ∈ I , and lim x = x. We will let S (A, x) be the subset of S(A, x) of sequences { x } for which x > x for all n ∈ I and S (A, x) be the subset of S(A, x) of sequences {x } for which x < x for all n ∈ I . A ⊂R

+

n

n

n∈I

n

n→∞

n

−

n

n

n∈I

n

n∈I

n

Definition Let D ⊂ R, f

: D → R, L ∈ R,

and suppose a is a limit point of D. We say the limit of f as x approaches a is L, denoted lim f (x) = L,

(5.1.1)

x→a

if for every sequence {x

n }n∈I

∈ S(D, a)

, lim f (xn ) = L.

(5.1.2)

n→∞

If S

+

(D, a) ≠ ∅,

we say the limit from the right of f as x approaches a is L, denoted lim f (x) = L,

(5.1.3)

x→a+

if for every sequence {x

n }n∈I

∈ S

+

(D, a)

, lim f (xn ) = L,

(5.1.4)

n→∞

and, if S

−

(D, a) ≠ ∅,

we say the limit from the left of f as x approaches a is L, denoted lim f (x) = L,

(5.1.5)

x→a−

if for every sequence {x

n }n∈I

∈ S

−

(D, a)

, lim f (xn ) = L.

(5.1.6)

lim f (x) = L

(5.1.7)

n→∞

We may also denote x→a

by writing f (x) → L as x → a.

(5.1.8)

Similarly, we may denote lim f (x) = L

(5.1.9)

+

x→a

by writing f (x) → L as x ↓ a

(5.1.10)

lim f (x) = L

(5.1.11)

and −

x→a

by writing f (x) → L as x ↑ a

(5.1.12)

f (a+) = lim f (x)

(5.1.13)

We also let +

x→a

and

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f (a−) = lim f (x).

(5.1.14)

−

x→a

It should be clear that if lim S (D, a) ≠ ∅, then f (a−) = L .

x→a

f (x) = L

and

S

+

(D, a) ≠ ∅,

then

f (a+) = L

. Similarly, if

and

limx→a f (x) = L

−

Proposition 5.1.1 Suppose D ⊂ R, f

and a is a limit point of D. If f (a−) = f (a+) = L, then lim

: D → R,

x→a

f (x) = L

.

Proof Suppose {x

∞

n }n=m

∈ S(D, a).

Let J

−

= {n : n ∈ Z, xn < a}

(5.1.15)

= {n : n ∈ Z, xn > a} .

(5.1.16)

and J

Suppose

J

is empty or finite and let ∈ S (D, a), and so

−

∞

+

k = m −1

if

J

−

=∅

and, otherwise, let

k

be the largest integer in

J

−

.

Then

+

{ xn }

n=k+1

lim f (xn ) = f (a+) = L.

(5.1.17)

n→∞

A similar argument shows that if J

is empty or finite, then

+

lim f (xn ) = f (a−) = L.

(5.1.18)

n→∞

If neither { xn }

n∈J

J

−

∈ S

−

nor −

J

+

(D, a)

is finite or empty, then {x } − and {x } + are subsequences of {x } with and {x } ∈ S (D, a). Hence, given any ϵ > 0, we may find integers N and M such that ∞

n

n

n∈J

n

n∈J

n

n∈J+

|f (xn ) − L| < ϵ

whenever n ∈ {j : j ∈ J

−

, j > N}

(5.1.19)

and |f (xn ) − L| < ϵ

whenever n ∈ {j : j ∈ J follows that

n=m

+

+

, j > M} .

Let

P

be the larger of

N

and

(5.1.20)

M.

Since

J

−

∪J

|f (xn ) − L| < ϵ

whenever n > P . Hence lim

n→∞

f (xn ) = L,

and so lim

x→a

f (x) = L

+

+

= {j : j ∈ Z

, j ≥ m} ,

it

(5.1.21)

.

Q.E.D.

Proposition 5.1.2 Suppose D ⊂ R, a is a limit point of D, and f

: D → R

. If lim

x→a

f (x) = L

and α ∈ R, then

lim αf (x) = αL.

(5.1.22)

x→a

Proof Suppose {x

n }n∈I

∈ S(D, a).

Then lim αf (xn ) = α lim f (xn ) = αL.

n→∞

Hence lim

x→a

αf (x) = αL

.

(5.1.23)

n→∞

Q.E.D.

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Proposition 5.1.3 Suppose D ⊂ R, a is a limit point of D, f

: D → R,

and g : D → R. If lim

x→a

f (x) = L

and lim

x→a

g(x) = M ,

lim(f (x) + g(x)) = L + M .

then (5.1.24)

x→a

Proof Suppose {x

n }n∈I

∈ S(D, a).

Then lim (f (xn ) + g (xn )) = lim f (xn ) + lim g (xn ) = L + M .

n→∞

Hence lim

x→a

n→∞

(f (x) + g(x)) = L + M

.

(5.1.25)

n→∞

Q.E.D.

Proposition 5.1.4 Suppose D ⊂ R, a is a limit point of D, f

: D → R,

and g : D → R. If lim

x→a

f (x) = L

and lim

x→a

g(x) = M ,

lim f (x)g(x) = LM .

then (5.1.26)

x→a

Exercise 5.1.1 Prove the previous proposition.

Proposition 5.1.5 Suppose

D ⊂ R, a

is

a

limit

limx→a f (x) = L, limx→a g(x) = M ,

point of D, f : D → R , and M ≠ 0, then f (x) lim x→a

g : D → R,

and

g(x) ≠ 0

for

all

x ∈ D.

If

L =

g(x)

.

(5.1.27)

M

Exercise 5.1.2 Prove the previous proposition.

Proposition 5.1.6 Suppose D ⊂ R, a is a limit point of D, f

: D → R,

and f (x) ≥ 0 for all x ∈ D. If lim

x→a

− − − − − − lim √f (x) = √L .

f (x) = L,

then (5.1.28)

x→a

Exercise 5.1.3 Prove the previous proposition. Given D ⊂ R, f

: D → R,

and A ⊂ D, we let f (A) = {y : y = f (x) for some x ∈ A}.

(5.1.29)

In particular, f (D) denotes the range of f .

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Proposition 5.1.7 Suppose

is a limit point of D, g : D → R , f : E → R, and and, for some ϵ > 0 , g(x) ≠ b for all x ∈ (a − ϵ, a + ϵ) ∩ D. If lim

D ⊂ R, E ⊂ R, a

limx→a g(x) = b

x→b

g(D) ⊂ E. f (x) = L,

Moreover, then

lim f ∘ g(x) = L.

suppose

(5.1.30)

x→a

Proof Suppose {x

n }n∈I

∈ S(D, a).

Then lim g (xn ) = b.

(5.1.31)

n→∞

Let N

+

∈ Z

such that |x

n

− a| < ϵ

whenever n > N . Then {g (xn )}

∞ n=N +1

∈ S(E, b),

(5.1.32)

so lim f (g (xn )) = L.

(5.1.33)

n→∞

Thus lim

x→a

f ∘ g(x) = L

.

Q.E.D.

Example 5.1.1 Let 0,

if x ≠ 0,

1,

if x = 0.

g(x) = {

(5.1.34)

If f (x) = g(x), then 1,

if x ≠ 0,

0,

if x = 0.

f ∘ g(x) = {

Hence lim

x→0

f ∘ g(x) = 1,

although lim

x→0

g(x) = 0

(5.1.35)

and lim

x→0

f (x) = 0

.

5.1.1 Limits of Polynomials and Rational Functions Example 5.1.2 If c ∈ R and f

: R → R

is given by f (x) = c for all x ∈ R, then clearly lim

x→a

f (x) = c

for any a ∈ R .

Example 5.1.3 Suppose f

: R → R

is defined by f (x) = x for all x ∈ R. If, for any a ∈ R, {x

n }n∈I

lim f (xn ) = lim xn = a.

n→∞

Hence lim

x→a

x =a

∈ S(R, a),

then (5.1.36)

n→∞

.

Example 5.1.4 Suppose n ∈ Z

+

and f

: R → R

is defined by f (x) = x . Then n

n n

lim f (x) = lim x x→a

n

= ∏ lim x = a .

x→a

(5.1.37)

x→a i=1

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Definition If n ∈ Z, n ≥ 0, and b

0,

b1 , … , bn

are real numbers with b

≠ 0,

n

n

p(x) = bn x

then we call the function p : R → R defined by

n−1

+ bn−1 x

+ ⋯ + b1 x + b0

(5.1.38)

a polynomial of degree n .

Exercise 5.1.4 Show that if f is a polynomial and a ∈ R, then lim

x→a

f (x) = f (a)

.

Definition Suppose p and q are polynomials and D = {x : x ∈ R, q(x) ≠ 0}.

(5.1.39)

We call the function r : D → R defined by p(x) r(x) =

(5.1.40) q(x)

a rational function.

Exercise 5.1.5 Show that if f is a rational function and a is in the domain of f , then lim

x→a

f (x) = f (a)

.

Exercise 5.1.6 Suppose

is a limit point of D, and lim for all x ∈ E, show that lim g(x) = L.

D ⊂ R, a ∈ D

g(x) = f (x)

x→a

f (x) = L

. If

E = D∖{a}

and

g : E → R

is defined by

x→a

Exercise 5.1.7 Evaluate 5

x lim x→1

3

x

−1 .

(5.1.41)

−1

Exercise 5.1.8 Suppose

is a limit point of D, f : D → R, g : D → R , h : D → R, and f (x) ≤ h(x) ≤ g(x) for all x ∈ D. If and lim g(x) = L, show that lim h(x) = L. (This is the squeeze theorem for limits of functions.)

D ⊂ R, a

limx→a f (x) = L

x→a

x→a

Note that the above results which have been stated for limits will hold as well for the appropriate one-sided limits, that is, limits from the right or from the left.

Exercise 5.1.9 Suppose ⎧ x + 1, ⎪ f (x) = ⎨ 4, ⎩ ⎪

Evaluate f (0), f (0−), and f (0+). Does lim

x→0

f (x)

2

x ,

if x < 0, if x = 0,

(5.1.42)

if x > 0.

exist?

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5.1.2 Equivalent Definitions Proposition 5.1.8 Suppose D ⊂ R, a is a limit point of D, and f δ > 0 such that

: D → R

. Then lim

x→a

f (x) = L

if and only if for every ϵ > 0 there exists a

|f (x) − L| < ϵ whenever x ≠ a and x ∈ (a − δ, a + δ) ∩ D.

(5.1.43)

Proof Suppose

limx→a f (x) = L.

x ∈ (a − δ, a + δ) ∩ D, x ≠ a,

Suppose there exists an ϵ > 0 such that for every for which |f (x) − L| ≥ ϵ . For n = 1, 2, 3, … , choose 1 xn ∈ (a −

such that |f (x ) − L| ≥ ϵ. Then {x assumption that lim f (x) = L . xn ≠ a,

∞

n }n=1

n

δ >0

there

exists

1 ,a+

n

∈ S(D, a),

) ∩ D,

(5.1.44)

n

but {f (x

∞

n )}n=1

does not converge to L, contradicting the

x→a

Now suppose that for every ϵ > 0 there exists δ > 0 such that |f (x) − L| < ϵ whenever x ≠ a and x ∈ (a − δ, a + δ) ∩ D. Let {x } ∈ S(D, a). Given ϵ > 0 , let δ > 0 be such that |f (x) − L| < ϵ whenever x ≠ a and x ∈ (a − δ, a + δ) ∩ D. Choose N ∈ Z such that |x − a| < δ whenever n > N . Then |f (x ) − L| < ϵ for all n > N . Hence lim f (x ) = L, and so lim f (x) = L. Q.E.D. n

n∈I

n

n→∞

n

n

x→a

The proofs of the next two propositions are analogous.

Proposition 5.1.9 Suppose D ⊂ R, a is a limit point of ϵ > 0 there exists a δ > 0 such that

D, f : D → R,

and

S

−

(D, a) ≠ ∅.

Then

limx→a− f (x) = L

|f (x) − L| < ϵ whenever x ∈ (a − δ, a) ∩ D.

if and only if for every

(5.1.45)

Proposition 5.1.10 Suppose D ⊂ R, a is a limit point of ϵ > 0 there exists a δ > 0 such that

D, f : D → R,

and

S

+

(D, a) ≠ ∅.

Then

limx→a+ f (x) = L

|f (x) − L| < ϵ whenever x ∈ (a, a + δ) ∩ D.

if and only if for every

(5.1.46)

5.1.3 Examples

Example 5.1.5 Define f

: R → R

by 1,

if x is rational,

0,

if x is irrational.

f (x) = {

(5.1.47)

Let a ∈ R. Since every open interval contains both rational and irrational numbers, for any there will exist x ∈ (a − δ, a + δ), x ≠ a, such that

δ >0

and any choice of

L ∈ R,

1 |f (x) − L| ≥

.

(5.1.48)

2

Hence lim

x→a

f (x)

does not exist for any real number a .

5.1.6

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Example 5.1.6 Define f

: R → R

by f (x) = {

Then lim

x→0

f (x) = 0

x,

if x is rational,

0,

if x is irrational.

(5.1.49)

since, given ϵ > 0, |f (x)| < ϵ provided |x| < ϵ.

Exercise 5.1.9 Show that if f is as given in the previous example and a ≠ 0 , then lim

x→a

does not exist.

f (x)

Exercise 5.1.11 Define f

: R → R

by 1

f (x) = {

q

,

if x is rational and x =

0,

p q

, (5.1.50)

if x is irrational,

where p and q are taken to be relatively prime integers with number a, lim f (x) = 0 .

and we take

q > 0,

q =1

when

x = 0.

Show that, for any real

x→a

Example 5.1.7 Define φ : [0, 1] → [−1, 1] by ⎧ 4x, ⎪ ⎪

if 0 ≤ x ≤

φ(x) = ⎨ 2 − 4x, ⎪ ⎩ ⎪

4x − 4,

if if

1 4 3 4

0, σ((0, ϵ)) = [−1, 1], and so lim σ(x) exist.

]) = s([n, n + 1]) = [−1, 1].

(5.1.54)

n limx→0+ σ(x)

does not exist. Similarly, neither

limx→0− σ(x)

nor

x→0

Example 5.1.8 Let s be the sawtooth function of the previous example and let D = R∖{0}. Define ψ : D → R by 1 ψ(x) = xs (

).

(5.1.55)

x

See Figure 5.1 .2 for the graph of ψ. Then for all x ∈ D, −|x| ≤ ψ(x) ≤ |x|,

and so lim

x→0

ψ(x) = 0

(5.1.56)

by the squeeze theorem.

Definition Let D ⊂ R and f

: D → R.

We say f is bounded if there exists a real number B such that |f (x)| ≤ B for all x ∈ D.

Exercise 5.1.12 Suppose f

: R → R

is bounded. Show that lim

x→0

xf (x) = 0

.

This page titled 5.1: Limits is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

5.1.8

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5.2: Monotonic Functions Definition Suppose D ⊂ R, f : D → R, and (a, b) ⊂ D. We say f is increasing on (a, b) if f (x) < f (y) whenever a < x < y < b; we say f is decreasing on (a, b) if f (x) > f (y) whenever a < x < y < b; we say f is nondecreasing on (a, b) if f (x) ≤ f (y) whenever a < x < y < b; and we say f is nonincreasing on (a, b) if f (x) ≥ f (y) whenever a < x < y < b. We will say f is monotonic on (a, b) if f is either nondecreasing or nonincreasing on (a, b) and we will say f is strictly monotonic on (a, b) if f is either increasing or decreasing on (a, b) .

Proposition 5.2.1 If f is monotonic on (a, b), then f (c+) and f (c−) exist for every c ∈ (a, b) . Proof Suppose f is nondecreasing on (a, b). Let c ∈ (a, b) and let λ = sup{f (x) : a < x < c}.

(5.2.1)

Note that λ ≤ f (c) < +∞. Given any ϵ > 0, there must exist δ > 0 such that λ − ϵ < f (c − δ) ≤ λ.

(5.2.2)

|f (x) − λ| < ϵ

(5.2.3)

Since f is nondecreasing, it follows that

whenever x ∈ (c − δ, c). Thus f (c−) = λ. A similar argument shows that f (c+) = κ where κ = inf{f (x) : c < x < b}.

(5.2.4)

f (c−) = inf{f (x) : a < x < c}

(5.2.5)

f (c+) = sup{f (x) : c < x < b}.

(5.2.6)

If f is nonincreasing, similar arguments yield

and

Proposition 5.2.2 If f is nondecreasing on (a, b) and a < x < y < b, then f (x+) ≤ f (y−).

(5.2.7)

Proof By the previous proposition, f (x+) = inf{f (t) : x < t < b}

(5.2.8)

f (y−) = sup{f (t) : a < t < y}.

(5.2.9)

and

Since f is nondecreasing, inf{f (t) : x < t < b} = inf{f (t) : x < t < y}

(5.2.10)

sup{f (t) : a < t < y} = sup{f (t) : x < t < y}.

(5.2.11)

and

5.2.1

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Thus f (x+) = inf{f (t) : x < t < y} ≤ sup{f (t) : x < t < y} = f (y−).

(5.2.12)

Q.E.D.

Exercise 5.2.1 Let φ : Q ∩ [0, 1] → Z

+

be a one-to-one correspondence. Define f

: [0, 1] → R

by

1 f (x) =

∑

φ(q)

q∈Q∩[0,1]

.

(5.2.13)

2

q≤x

a. Show that f is increasing on (0, 1). b. Show that for any x ∈ Q ∩ (0, 1), f (x−) < f (x) and f (x+) = f (x). c. Show that for any irrational a, 0 < a < 1, lim

x→a

f (x) = f (a)

.

This page titled 5.2: Monotonic Functions is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

5.2.2

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5.3: Limits to Infinity and Infinite Limits Definition Let D ⊂ R, f

and suppose a is a limit point of D. We say that f diverges to +∞ as x approaches a , denoted

: D → R,

lim f (x) = +∞,

(5.3.1)

x→a

if for every real number M there exists a δ > 0 such that f (x) > M whenever x ≠ a and x ∈ (a − δ, a + δ) ∩ D.

(5.3.2)

Similarly, we say that that f diverges to −∞ as x approaches a, denoted lim f (x) = −∞,

(5.3.3)

x→a

if for every real number M there exists a δ > 0 such that f (x) < M whenever x ≠ a and x ∈ (a − δ, a + δ) ∩ D.

(5.3.4)

Exercise 5.3.1 Provide definitions for a. lim

f (x) = +∞

−

f (x) = +∞

+

f (x) = −∞

−

f (x) = −∞

b. lim

x→a

c. lim

x→a

d. lim

,

+

x→a

x→a

,

, .

Model your definitions on the preceding definitions.

Exercise 5.3.2 Show that lim

7 +

x→4

4−x

= −∞

and lim

7 −

x→4

4−x

= +∞

.

Definition Suppose D ⊂ R does not have an upper bound, f denoted

: D → R

, and L ∈ R. We say that the limit of f as x approaches +∞ is L,

lim

f (x) = L,

(5.3.5)

x→+∞

if for every ϵ > 0 there exists a real number M such that |f (x) − L| < ϵ whenever x ∈ (M , +∞) ∩ D.

(5.3.6)

Definition Suppose D ⊂ R does not have an lower bound, f denoted

: D → R

, and L ∈ R. We say that the limit of f as x approaches −∞ is L,

lim

f (x) = L,

(5.3.7)

x→−∞

if for every ϵ > 0 there exists a real number M such that |f (x) − L| < ϵ whenever x ∈ (−∞, M ) ∩ D.

5.3.1

(5.3.8)

https://math.libretexts.org/@go/page/22666

Exercise 5.3.3 Verify that lim

x→+∞

x+1

=1

x+2

.

Exercise 5.3.4 Provide definitions for a. lim

f (x) = +∞

,

b. lim

x→+∞

f (x) = −∞

,

c. lim

x→−∞

f (x) = +∞

,

f (x) = −∞

.

x→+∞

d. lim

x→−∞

Model your definitions on the preceding definitions.

Exercise 5.3.5 Suppose 3

f (x) = ax

2

+ bx

+ cx + d,

(5.3.9)

where a, b, c, d ∈ R and a > 0. Show that lim x→+∞

f (x) = +∞ and

lim

f (x) = −∞.

(5.3.10)

x→−∞

This page titled 5.3: Limits to Infinity and Infinite Limits is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

5.3.2

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5.4: Continuous Functions 5.4.1 Continuity at a Point Definition Suppose

and a ∈ D. We say f is continuous at a if either a is an isolated point of If f is not continuous at a, we say f is discontinuous at a, or that f has a discontinuity at a.

D ⊂ R, f : D → R,

limx→a f (x) = f (a).

D

or

Example 5.4.1 Define f

: R → R

by 1,

if x is rational,

0,

if x is irrational.

f (x) = {

(5.4.1)

Then, by Example 5.1.5, f is discontinuous at every x ∈ R.

Example 5.4.2 Define f

: R → R

by x,

if x is rational,

0,

if x is irrational.

f (x) = {

(5.4.2)

Then, by Example 5.1 .6 and Exercise 5.1.10, f is continuous at 0, but discontinuous at every x ≠ 0. If D ⊂ R, α ∈ R, f

: D → R,

and g : D → R, then we define αf

: D → R

by

(αf )(x) = αf (x),

(5.4.3)

(f + g)(x) = f (x) + g(x),

(5.4.4)

(f g)(x) = f (x)g(x).

(5.4.5)

f + g : D → Rby

and f g : D → R by

Moreover, if g(x) ≠ 0 for all x ∈ D, we define

f g

by

: D → R

f (x)

f (

) (x) = g

.

(5.4.6)

g(x)

Proposition 5.4.1 Suppose

D ⊂ R, α ∈ R, f : D → R,

and

g : D → R.

If

continuous at a. Moreover, if g(x) ≠ 0 for all x ∈ D, then

and

f f g

g

are continuous at

a,

then

αf , f + g,

and

fg

are all

is continuous at a.

Exercise 5.4.1 Prove the previous proposition.

Proposition 5.4.2 Suppose

for all

D ⊂ R, f : D → R, f (x) ≥ 0 − − − − g(x) = √f (x), g a.

then

x ∈ D,

and

f

is continuous at

a ∈ D.

If

g : D → R

is defined by

is continuous at

5.4.1

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Exercise 5.4.2 Prove the previous proposition.

Proposition 5.4.3 Suppose D ⊂ R, f

: D → R,

and a ∈ D. Then f is continuous at a if and only if for every ϵ > 0 there exists δ > 0 such that |f (x) − f (a)| < ϵ whenever x ∈ (a − δ, a + δ) ∩ D.

(5.4.7)

Proof Suppose f is continuous at a . If a is an isolated point of D, then there exists a δ > 0 such that (a − δ, a + δ) ∩ D = {a}.

(5.4.8)

Then for any ϵ > 0, if x ∈ (a − δ, a + δ) ∩ D, then x = a, and so |f (x) − f (a)| = |f (a) − f (a)| = 0 < ϵ.

If a is a limit point of D, then lim

x→a

f (x) = f (a)

(5.4.9)

implies that for any ϵ > 0 there exists δ > 0 such that

|f (x) − f (a)| < ϵ whenever x ∈ (a − δ, a + δ) ∩ D.

(5.4.10)

Now suppose that for every ϵ > 0 there exists δ > 0 such that |f (x) − f (a)| < ϵ whenever x ∈ (a − δ, a + δ) ∩ D.

If a is an isolated point, then f is continuous at a . If and so f is continuous at a. Q.E.D. From the preceding, it should be clear that a function f {x } with x ∈ D for every n ∈ I and lim x n

n

n∈I

n→∞

n

a

(5.4.11)

is a limit point, then this condition implies

: D → R

limx→a f (x) = f (a),

is continuous at a point a of D if and only if for every sequence .

= a, limn→∞ f (xn ) = f (a)

Exercise 5.4.1 Show that if f : D → R is continuous at f (x) > 0 for every x ∈ I ∩ D.

a ∈ D

and

f (a) > 0

, then there exists an open interval

I

such that

a ∈ I

and

Proposition 5.4.4 Suppose D ⊂ R, E ⊂ R, g : D → R, f then f ∘ g is continuous at a.

: E → R, g(D) ⊂ E

and a ∈ D. If g is continuous at a and f is continuous at

g(a),

Proof Let

{ xn }

n∈I

a, {g (xn )}

at g(a),

be a sequence with x is a sequence with g (x

for every n ∈ I and lim for every n ∈ I and lim . That is,

n

n∈I

limn→∞ f (g (xn )) = f (g(a))

∈ D

n→∞

n) ∈ E

n→∞

lim (f ∘ g) (xn ) = (f ∘ g)(a).

xn = a

. Then, since g is continuous at Hence, since f is continuous

g (xn ) = g(a).

(5.4.12)

n→∞

Hence f ∘ g is continuous at a .

Definition Let D ⊂ R, f : D → R, and a ∈ D. If f is not continuous at a but both f (a−) and f (a+) exist, then we say f has a simple discontinuity at a.

5.4.2

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Proposition 5.4.5 Suppose f is monotonic on the interval (a, b). Then every discontinuity of f in (a, b) is a simple discontinuity. Moreover, if E is the set of points in (a, b) at which f is discontinuous, then either E = ∅, E is finite, or E is countable. Proof The first statement follows immediately from Proposition 5.2.1. For the second statement, suppose f is nondecreasing and suppose E is nonempty. From Exercise 2.1 .26 and the the proof of Proposition 5.2.1, it follows that for every x ∈ (a, b) , f (x−) ≤ f (x) ≤ f (x+).

Hence

x ∈ E

(5.4.13)

if and only if f (x−) < f (x+). Hence for every x ∈ E, we may choose a rational number Now if x, y ∈ E with x < y, then, by Proposition 5.2.2,

rx

such that

f (x−) < rx < f (x+).

rx < f (x+) ≤ f (y−) < ry ,

(5.4.14)

so r ≠ r . Thus we have a one-to-one correspondence between E and a subset of countable. A similar argument holds if f is nonincreasing. Q.E.D. x

y

Q,

and so

E

is either finite or

Exercise 5.4.4 Define f

: R → R

by 1

f (x) = {

q

p

,

0,

if x is rational and x =

q

, (5.4.15)

if x is irrational.

where p and q are taken to be relatively prime integers with q > 0, and we take q = 1 when x = 0. Show that f is continuous at every irrational number and has a simple discontinuity at every rational number.

5.4.2 Continuity on a Set Definition Suppose D ⊂ R and f

: D → R.

We say f is continuous on D if f is continuous at every point a ∈ D .

Proposition 5.4.6 If f is a polynomial, then f is continuous on R.

Proposition 5.4.7 If D ⊂ R and f

: D → R

is a rational function, then f is continuous on D.

Exercise 5.4.5 − −−− −

Explain why the function f (x) = √1 − x is continuous on [−1, 1]. 2

Exercise 5.4.6 Discuss the continuity of the function x + 1, ⎧ ⎪ f (x) = ⎨ 4, ⎩ ⎪

2

x ,

If D ⊂ R, f

: D → R,

if x < 0, if x = 0,

(5.4.16)

if x > 0.

and E ⊂ R, we let

5.4.3

https://math.libretexts.org/@go/page/22667

f

−1

(E) = {x : f (x) ∈ E}.

(5.4.17)

Proposition 5.4.8 Suppose D ⊂ R and f some open set U ⊂ R.

: D → R.

Then f is continuous on

D

if and only if for every open set

V ⊂ R, f

−1

(V ) = U ∩ D

for

Proof Suppose f is continuous on D and V ⊂ R is an open set. If V ∩ f (D) = ∅ , then f V ∩ f (D) ≠ ∅ and let a ∈ f (V ) . Since V is open and f (a) ∈ V , there exists ϵ

−1

−1

a

which is open. So suppose such that

(V ) = ∅,

>0

(f (a) − ϵa , f (a) + ϵa ) ⊂ V .

Since f is continuous, there exists δ

a

>0

(5.4.18)

such that

f ((a − δa , a + δa ) ∩ D) ⊂ (f (a) − ϵa , f (a) + ϵa ) ⊂ V .

That is, (a − δ

a,

a + δa ) ∩ D ⊂ f

−1

(V ).

Let U =

⋃ a∈f

Then U is open and f

−1

(5.4.19)

−1

(a − δa , a + δa ) .

(5.4.20)

(V )

.

(V ) = U ∩ D

Now suppose that for every open set V ⊂ R, f (V ) = U ∩ D for some open set given. Since (f (a) − ϵ, f (a) + ϵ) is open, there exists an open set U such that −1

U ∩D =f

−1

U ⊂ R.

Let

a ∈ D

and let

((f (a) − ϵ, f (a) + ϵ)).

ϵ>0

be

(5.4.21)

Since U is open and a ∈ U , there exists δ > 0 such that (a − δ, a + δ) ⊂ U . But then f ((a − δ, a + δ) ∩ D) ⊂ (f (a) − ϵ, f (a) + ϵ).

(5.4.22)

That is, if x ∈ (a − δ, a + δ) ∩ D, then |f (x) − f (a)| < ϵ. Hence f is continuous at a.

Q.E.D.

Exercise 5.4.7 Let D ⊂ R and f

: D → R.

For any E ⊂ R, show that f

−1

(R∖E) = (R∖ f

−1

(E)) ∩ D

.

Exercise 5.4.8 Let A be a set and, for each α ∈ A, let U

α

⊂ R.

Given D ⊂ R and a function f

⋃ f

−1

(Uα ) = f

−1

α∈A

: D → R,

show that

( ⋃ Uα )

(5.4.23)

α∈A

and ⋂ f

−1

(Uα ) = f

−1

α∈A

( ⋂ Uα ) .

(5.4.24)

α∈A

Exercise 5.4.9 Suppose D ⊂ R and f : D → R. Show that f is continuous on D if and only if for every closed set C for some closed set F ⊂ R.

5.4.4

⊂ R, f

−1

(C ) = F ∩ D

https://math.libretexts.org/@go/page/22667

Exercise 5.4.10 Let D ⊂ R. We say a function f : D → R is Lipschitz if there exists all x, y ∈ D. Show that if f is Lipschitz, then f is continuous.

α ∈ R, α > 0,

such that

|f (x) − f (y)| ≤ α|x − y|

for

5.4.3 Intermediate Value Theorem

Theorem 5.4.9 (Intermediate Value Theorem). Suppose

and f : [a, b] → R. If f is continuous and then there exists c ∈ [a, b] such that f (c) = s .

a, b ∈ R, a < b,

f (b) ≤ s ≤ f (a),

is such that either

s ∈ R

f (a) ≤ s ≤ f (b)

or

Proof Suppose f (a) < f (b) and f (a) < s < f (b). Let c = sup{x : x ∈ [a, b], f (x) ≤ s}.

Suppose

f (c) < s.

x ∈ (c, c + δ).

exists

δ >0

f (c) = s.

Then

But then

such that Q.E.D.

c 0

Similarly, if

c.

such that

f (c) > s,

again contradicting the definition of

x ∈ (c − δ, c),

c.

f (x) < s

then

c >a

for all and there

Hence we must have

Example 5.4.3 Suppose a ∈ R, a > 0, and consider f (x) = x

n

−a

where n ∈ Z, n > 1. Then f (0) = −a < 0 and n

f (1 + a) = (1 + a)

−a n

n

= 1 + na + ∑ (

i

) a −a i

i=2

n

n

= 1 + (n − 1)a + ∑ ( i=2

where (

n

)

i

)a

> 0,

i

is the binomial coefficient

i n (

n! ) =

.

i

(5.4.26)

i!(n − i)!

Hence, by the Intermediate Value Theorem, there exists a real number such γ since f is increasing on (0, +∞).

γ >0

such that

γ

n

= a.

Moreover, there is only one

We call γ the n th root of a, and write − n − γ = √a

(5.4.27)

or 1

γ =a

Moreover, if a ∈ R, a < 0, n ∈ Z

+

n

.

(5.4.28)

is odd, and γ is the nth root of −a, then n

(−γ )

n

n

= (−1 ) (γ )

= (−1)(−a) = a.

(5.4.29)

That is, −γ is the n th root of a .

5.4.5

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Definition If n =

p q

∈ Q

with q ∈ Z

+

,

then we define n

x

q − p = (√x )

(5.4.30)

for all real x ≥ 0 .

Exercise 5.4.11 Explain why the equation x

5

2

+ 4x

has a solution in the interval (0, 2).

− 16 = 0

Exercise 5.4.12 Give an example of a closed interval Intermediate Value Theorem.

[a, b] ⊂ R

and a function

f : [a, b] → R

which do not satisfy the conclusion of the

Exercise 5.4.13 Show that if I

⊂R

is an interval and f

: I → R

is continuous, then f (I ) is an interval.

Exercise 5.4.14 Suppose f : (a, b) → R is continuous and strictly monotonic. Let strictly monotonic and continuous.

(c, d) = f ((a, b)).

Show that

f

−1

: (c, d) → (a, b)

is

Exercise 5.4.15 Let n ∈ Z

+

.

Show that the function f (x) = √− x is continuous on (0, +∞). n

Exercise 5.4.16 Use the method of bisection to give another proof of the Intermediate Value Theorem. 5.4.4 Extreme Value Theorem

Theorem 5.4.10 Suppose D ⊂ R is compact and f

: D → R

is continuous. Then f (D) is compact.

Proof Given a sequence {y } in f (D), choose a sequence {x a convergent subsequence {x } with n

n }n∈I

n∈I

such that f (x

n)

= yn .

Since D is compact, {x

n }n∈I

has

∞

nk

k=1

lim xn

k

k→∞

= x ∈ D.

(5.4.31)

Let y = f (x). Then y ∈ f (D) and, since f is continuous, y = lim f (xn ) = lim yn . k→∞

k

k→∞

k

(5.4.32)

Hence f (D) is compact.

5.4.6

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Exercise 5.4.17 Prove the previous theorem using the open cover definition of a compact set.

Theorem 5.4.11 (Extreme Value Theorem). Suppose D ⊂ R is compact and f : D → R is continuous. Then there exists there exists b ∈ D such that f (b) ≤ f (x) for all x ∈ D. Q.E.D.

a ∈ D

such that

f (a) ≥ f (x)

for all

x ∈ D

and

As a consequence of the Extreme Value Theorem, a continuous function on a closed bounded interval attains both a maximum and a minimum value.

Exercise 5.4.18 Find an example of a closed bounded interval [a, b] and a function f minimum value on [a, b].

: [a, b] → R

such that f attains neither a maximum nor a

Exercise 5.4.19 Find an example of a bounded interval maximum nor a minimum value on I .

I

and a function

f : I → R

which is continuous on

I

such that

f

attains neither a

Exercise 5.4.20 Suppose K ⊂ R is compact and a ∉ K. Show that there exists b ∈ K such that |b − a| ≤ |x − a| for all x ∈ K .

Proposition 5.4.12 Suppose D ⊂ R is compact, f

: D → R

is continuous and one-to-one, and E = f (D). Then f

−1

: E → D

is continuous.

Proof Let V ⊂ R be an open set. We need to show that f (V ∩ D) = U ∩ E for some open set U ⊂ R . Let C = D ∩ (R∖V ). Then C is a closed subset of D, and so is compact. Hence f (C ) is a compact subset of E. Thus f (C ) is closed, and so U = R∖f (C ) is open. Moreover, U ∩ E = E∖f (C ) = f (V ∩ D). Thus f is continuous. −1

Exercise 5.4.21 Suppose f

: [0, 1] ∪ (2, 3] → [0, 2]

by x,

if 0 ≤ x ≤ 1,

x − 1,

if 2 < x ≤ 3.

f (x) = {

(5.4.33)

Show that f is continuous, one-to-one, and onto, but that f

−1

is not continuous.

5.4.5 Uniform Continuity

Definition Suppose D ⊂ R and f any x, y ∈ D,

: D → R.

We say

f

is uniformly continuous on

D

if for every

|f (x) − f (y)| < ϵ whenever |x − y| < δ.

5.4.7

ϵ>0

there exists

δ >0

such that for

(5.4.34)

https://math.libretexts.org/@go/page/22667

Exercise 5.4.22 Suppose D ⊂ R and f

: D → R

is Lipschitz (see Exercise 5.4.10). Show that f is uniformly continuous on D.

Clearly, if f is uniformly continuous on continuous.

D

then

f

is continuous on

However, a continuous function need not be uniformly

D.

Example 5.4.4 Define f

: (0, +∞)

by f (x) =

1 x

,

Given any δ > 0, choose n ∈ Z

+

1 |x − y| =

such that

1

< δ.

Let x =

1 n

and y =

1 n+1

.

Then

1

− n

1 n(n+1)

= n+1

< δ.

(5.4.35)

n(n + 1)

However, |f (x) − f (y)| = |n − (n + 1)| = 1.

(5.4.36)

Hence, for example, there does not exist a δ > 0 such that 1 |f (x) − f (y)|

0 be given. If δ =

ε 2

, then

|f (x) − f (y)| = 2|x − y| < ϵ

(5.4.38)

whenever |x − y| < δ. Hence f is uniformly continuous on R.

Exercise 5.4.23 Let f (x) = x

2

.

Show that f is not uniformly continuous on (−∞, +∞) .

Proposition 5.4.13 Suppose D ⊂ R is compact and f

: D → R

is continuous. Then f is uniformly continuous on D.

Proof Let ϵ > 0 be given. For every x ∈ D, choose δ such that x

ϵ |f (x) − f (y)|

0 ,

Exercise 6.3.7 Suppose I is an open interval, f all x ∈ I .

: I → R,

and f

′

(x) = 0

for all

x ∈ I.

Show that there exists α ∈ R such that

f (x) = α

for

Exercise 6.3.8 Suppose I is an open interval, f

: I → R, g : I → R,

and f

′

′

(x) = g (x)

for all x ∈ I . Show that there exists α ∈ R such that

g(x) = f (x) + α

(6.3.14)

for all x ∈ I .

Exercise 6.3.9 Let D = R∖{0}. Define f

: D → R

and g : D → R by f (x) = x and 2

2

x , g(x) = {

2

x

(6.3.15) + 1,

Show that f (x) = g (x) for all x ∈ D, but there does not exist this not contradict the conclusion of the previous exercise? ′

′

if x < 0, if x > 0.

α ∈ R

such that

g(x) = f (x) + α

for all

x ∈ D.

Why does

Proposition 6.3.5 If f is differentiable on (a, b) and f

′

(x) > 0

for all x ∈ (a, b), then f is increasing on (a, b).

Proof Let x, y ∈ (a, b) with x < y. By the Mean Value Theorem, there exists a point c ∈ (x, y) such that ′

f (y) − f (x) = (y − x)f (c).

Since y − x > 0 and f

′

(c) > 0,

we have f (y) > f (x), and so f is increasing on (a, b).

6.3.3

(6.3.16)

Q.E.D.

https://math.libretexts.org/@go/page/22673

Theorem 6.3.6 If f is differentiable on (a, b) and f

′

(x) < 0

for all x ∈ (a, b), then f is decreasing on (a, b).

Exercise 6.3.10 State and prove similar conditions for nonincreasing and nondecreasing functions. This page titled 6.3: Mean Value Theorem is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

6.3.4

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6.4: Discontinuities of Derivatives Theorem 6.4.1: Intermediate Value Theorem for Derivatives Suppose f

′

is differentiable on an open interval I , a, b ∈ I , and (a) > λ > f (b), then there exists c ∈ (a, b) such that f (c) = λ . f

′

a < b.

If

λ ∈ R

and either

′

′

f (a) < λ < f (b)

or

′

Proof Suppose f (a) < λ < f (b) and define g : I → R by g(x) = f (x) − λx . Then g is differentiable on continuous on [a, b]. Let c be a point in [a, b] at which g attains its minimum value. Now ′

′

′

′

I,

and so

g (a) = f (a) − λ < 0,

(6.4.1)

g(t) − g(a) < 0.

(6.4.2)

so there exists a < t < b such that

Thus c ≠ a. Similarly, ′

′

g (b) = f (b) − λ > 0,

(6.4.3)

g(s) − g(b) < 0.

(6.4.4)

so there exists a < s < b such that

Thus c ≠ b. Hence c ∈ (a, b), and so g

′

(c) = 0.

Thus 0 = f

′

(c) − λ,

and so f

′

(c) = λ.

Q.E.D.

Exercise 6.4.1 Define g : (−1, 1) → R by −1,

if − 1 < x < 0,

1,

if 0 ≤ x < 1.

g(x) = {

Does there exist a function f

: (−1, 1) → R

such that f

(6.4.5)

′

for all x ∈ (−1, 1)?

(x) = g(x)

Exercise 6.4.2 Suppose f is differentiable on an open interval I . Show that f cannot have any simple discontinuities in I . ′

Example 6.4.1 Define φ : [0, 1] → R by φ(x) = x(2x − 1)(x − 1). Define ρ : R → R by ρ(x) = 6x

2

3

φ(x) = 2 x

2

− 3x

− 6x + 1.

+ x,

Then (6.4.6)

so φ (x) = ρ(x) for all x ∈ (0, 1). Next define s : R → R by s(x) = φ(x − ⌊x⌋) . See Figure 6.4.1 for the graphs of φ and s. Then for any n ∈ Z and n < x < n + 1 , ′

′

s (x) = ρ(x − n) = ρ(x − ⌊x⌋).

(6.4.7)

Moreover, if x is an integer, s(x + h) − s(x) lim +

h→0

φ(h) = lim

h

+

h→0

h h(2h − 1)(h − 1)

= lim +

h→0

h

= lim (2h − 1)(h − 1) +

h→0

=1

6.4.1

https://math.libretexts.org/@go/page/22674

and s(x + h) − s(x)

φ(h + 1)

lim

= lim h

−

h→0

h

−

h→0

(h + 1)(2h + 1)h = lim h

−

h→0

= lim (h + 1)(2h + 1) −

h→0

= 1.

Figure 6.4.1 : Graphs of y = φ(x) and y = s(x)

Thus s (x) = 1 = ρ(x − ⌊x⌋) when x is an integer, and so s (x) = ρ(x − ⌊x⌋) for all x ∈ R. ′

Now

′

ρ(x) = 0

φ(1) = 0,

if and only if

x =

3−√3 6

or

x =

3+√3 6

since

.

φ(0) = 0

,

φ(

3−√3 6

1

) =

,φ(

6 √3

3+√3 6

) =−

1

and

,

6 √3

we see that φ attains a

maximum value of

1 6 √3

and a minimum value of −

1

.

6 √3

Hence for any n ∈ Z , 1

s((n, n + 1)) = [−

Also, ρ (x) = 12x − 6, so ρ (x) = 0 if and only if x = maximum value of 1 and a v, ′

′

1 2

1

– 6 √3

,

– 6 √3

].

since ρ(0) = 1 , ρ (

.

1 2

(6.4.8)

) =−

1 2

,

and ρ(1) = 1, we see that ρ attains a

1

′

s ((n, n + 1)) = [−

, 1] .

(6.4.9)

2

It follows from the preceding, in the same manner as the result in Example the function g(x) = s ( ) has a limit as x approaches 0. ′

5.1.7,

that neither the function

σ(x) = s (

1 x

)

nor

1

x

Finally, define ψ : R → R by 2

x s( ψ(x) = {

1 x

),

if x ≠ 0, (6.4.10)

0,

if x = 0.

For x ≠ 0, we have ′

2

′

1

ψ (x) = x s (

1 ) (−

x

2

1 ) + 2xs (

′

1

) = −s ( x

x

1 ) + 2xs (

x

).

(6.4.11)

x

At 0, we have

6.4.2

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ψ(0 + h) − ψ(0)

′

ψ (0) = lim h

h→0 2

h s( = lim

1 h

)

h

h→0

1 = lim hs (

) h

h→0

= 0,

where the final limit follows from the squeeze theorem and the fact that s is bounded. Hence we see that ψ is continuous on R and differentiable on R, but ψ is not continuous since ψ (x) does not have a limit as x approaches 0. See Figure 6.4.1 for the graphs of ψ and ψ . ′

′

′

Figure 6.4.2 : Graphs of y = ψ(x) and y = ψ (x) . ′

Exercise 6.4.3 Let s be as above and define g : R → R by 4

x s( g(x) = {

1 x

),

0,

if x ≠ 0 (6.4.12) if x = 0.

Show that g is differentiable on R and that g is continuous on R. ′

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6.4.3

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6.5: L'Hopital's Rule The following result is one case of l l'Hópital's rule. ′

Theorem 6.5.1 Suppose a, b ∈ R, f and g are differentiable on (a, b), g

′

(x) ≠ 0

for all x ∈ (a, b), and

′

f (x) lim x→a

If lim

+

x→a

f (x) = 0

and lim

+

x→a

g(x) = 0,

= λ.

′

(6.5.1)

g (x)

+

then f (x) lim

= λ.

+

(6.5.2)

g(x)

x→a

Proof Given ϵ > 0, there exists δ > 0 such that ′

f (x)

ϵ λ−

< 2

ϵ 0. (1)

′

′′

′′

c.

Show that

f

has a local

′′

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6.6.3

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CHAPTER OVERVIEW 7: Integrals 7.1: Upper and Lower Integrals 7.2: Integrals 7.3: Integrability Conditions 7.4: Properties of Integrals 7.5: The Fundamental Theorem of Calculus 7.6: Taylor's Theorem Revisited 7.7: An Improper Integral

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1

7.1: Upper and Lower Integrals Definition Given a closed interval [a, b].

[a, b] ⊂ R

with

a < b,

we call any finite subset of

For convenience, whenever we consider a partition P of an interval with 0. That is, if |P | = n + 1 and P = {x , x , … , x } , then 0

1

which includes both

[a, b]

a

and

b

a partition of

we will index the elements in increasing order, starting

[a, b]

n

a = x0 < x1 < x2 < ⋯ < xn = b.

(7.1.1)

Definition Suppose P

= { x0 , x1 , … , xn }

is a partition of [a, b] and f

is bounded. For i = 1, 2, … , n, let

: [a, b] → R

mi = inf {f (x) : xi−1 ≤ x ≤ xi }

(7.1.2)

Mi = sup {f (x) : xi−1 ≤ x ≤ xi } .

(7.1.3)

and

We call n

L(f , P ) = ∑ mi (xi − xi−1 )

(7.1.4)

i=1

the lower sum of f determined by P and n

U (f , P ) = ∑ Mi (xi − xi−1 )

(7.1.5)

i=1

the upper sum of f determined by P .

Definition If P and P are both partitions of [a, b] and P 1

2

1

⊂ P2 ,

then we call P a refinement of P . 2

1

Definition If P and P are both partitions of [a, b], then we call the partition P 1

2

= P1 ∪ P2

the common refinement of P and P . 1

2

lemma 7.1.1 Suppose P = {x , x , … , x } is a partition of [a, b], s ∈ (a, b) , s ∉ P then L (f , P ) ≤ L (f , P ) and U (f , P ) ≤ U (f , P ) . 1

0

1

1

1,

n

2

2

and f

: [a, b] → R

is bounded. If

P2 = P1 ∪ {s},

1

Proof Suppose x

i−1

< s < xi

and let w1 W1 w2 W2 mi

= inf {f (x) : xi−1 ≤ x ≤ s} , = sup {f (x) : xi−1 ≤ x ≤ s} , = inf {f (x) : s ≤ x ≤ xi } , = sup {f (x) : s ≤ x ≤ xi } , = inf {f (x) : xi−1 ≤ x ≤ xi } ,

and

7.1.1

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Mi = sup {f (x) : xi−1 ≤ x ≤ xi } .

Then w

1

and W

≥ mi , w2 ≥ mi , W1 ≤ Mi ,

2

L (f , P2 ) − L (f , P1 )

≤ Mi .

(7.1.6)

Hence

= w1 (s − xi−1 ) + w2 (xi − s) − mi (xi − xi−1 ) = w1 (s − xi−1 ) + w2 (xi − s) − mi (s − xi−1 ) − mi (xi − s) = (w1 − mi ) (s − xi−1 ) + (w2 − mi ) (xi − s) ≥0

and U (f , P1 ) − U (f , P2 ) = Mi (xi − xi−1 ) − W1 (s − xi−1 ) − W2 (xi − s) = Mi (s − xi−1 ) + Mi (xi − s) − W1 (s − xi−1 ) − W2 (xi − s) = (Mi − W1 ) (s − xi−1 ) + (Mi − W2 ) (xi − s) ≥0.

Thus L (f , P

1)

≤ L (f , P2 )

and U (f , P

2)

.

≤ U (f , P1 )

Q.E.D.

Proposition 7.1.2 Suppose

P1

and

P2

L (f , P1 ) ≤ L (f , P2 )

are partitions of [a, b], with and U (f , P ) ≤ U (f , P ) . 2

P2

a refinement of

P1 .

If

f : [a, b] → R

is bounded, then

1

Proof The proposition follows immediately from repeated use of the previous lemma.

Q.E.D.

Proposition 7.1.3 Suppose P and P are partitions of [a, b]. If f 1

2

: [a, b] → R

is bounded, then L (f , P

1)

≤ U (f , P2 )

.

Proof The result follows immediately from the definitions if P Then

1

= P2 .

Otherwise, let P be the common refinement of P and P

L (f , P1 ) ≤ L(f , P ) ≤ U (f , P ) ≤ U (f , P2 ) .

1

2.

(7.1.7)

Q.E.D.

Definition Suppose a < b and f

: [a, b] → R

is bounded. We call b

∫

f = sup{L(f , P ) : P is a partition of [a, b]}

(7.1.8)

a

––––

the lower integral of f over [a, b] and ¯ ¯¯¯¯¯¯ ¯ b

∫

f = inf{U (f , P ) : P is a partition of [a, b]}

(7.1.9)

a

the upper integral of f over [a, b]. Note that both the lower integral and the upper integral are finite real numbers since the lower sums are all bounded above by any upper sum and the upper sums are all bounded below by any lower sum.

7.1.2

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Proposition 7.1.4 Suppose a < b and f

: [a, b] → R

is bounded. Then ¯ ¯¯¯¯¯¯ ¯ b

b

∫

f ≤∫

a

f.

(7.1.10)

a

––––

Proof Let P be a partition of [a, b]. Then for any partition Q of [a, b], we have L(f , Q) ≤ U (f , P ). Hence U (f , P ) is an upper bound for any lower sum, and so b

∫

f ≤ U (f , P ).

(7.1.11)

a

––––

But this shows that the lower integral is a lower bound for any upper sum. Hence b

∫

b

f ≤∫

a

f.

(7.1.12)

a

Q.E.D.

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7.2: Integrals Definition Suppose a < b and f

: [a, b] → R

is bounded. We say f is ¯ ¯¯¯¯¯¯ ¯ b

b

∫

f =∫

a

f.

(7.2.1)

a

––––

If f is integrable, we call the common value of the upper and lower integrals the integral of f over [a, b], denoted b

∫

f.

(7.2.2)

a

That is, if f is integrable on [a, b], b

∫

¯ ¯¯¯¯¯¯ ¯ b

b

f = ∫

a

f =∫

a

f.

(7.2.3)

a

––––

Example 7.2.1 Define f

: [0, 1] → R

by 1,

if x ∈ Q,

0,

if x ∉ Q.

f (x) = {

For any partition P

= { x0 , x1 , … , xn } ,

(7.2.4)

we have n

L(f , P ) = ∑ 0 (xi − xi−1 ) = 0

(7.2.5)

i=1

and n

U (f , P ) = ∑ (xi − xi−1 ) = xn − x0 = 1.

(7.2.6)

i=1

Thus 1

∫

f =0

(7.2.7)

f = 1.

(7.2.8)

0

––––

and ¯ ¯¯¯¯¯¯ ¯ 1

∫ 0

Hence f is not integrable on [0, 1].

Example 7.2.2 Define f

: [0, 1] → R

by 1

f (x) = {

q

p

,

0,

if x is rational and x =

q

, (7.2.9)

if x is irrational,

where p and q are taken to be relatively prime integers with q > 0, and we take q = 1 when x = 0. Show that on [0, 1] and

7.2.1

f

is integrable

https://math.libretexts.org/@go/page/22679

1

∫

f = 0.

(7.2.10)

0

Exercise 7.2.3 Let f

: [0, 1] → R

be defined by f (x) = x and, for n ∈ Z

+

let P

,

= { x0 , x1 , … , xn }

be the partition of [0, 1] with

i xi =

, i = 0, 1, … , n.

(7.2.11)

n

Show that 1 U (f , P ) − L(f , P ) =

,

(7.2.12)

n

and hence conclude that f is integrable on [0, 1]. Show that 1

1

∫

f =

.

(7.2.13)

2

0

Exercise 7.2.4 Define f

: [1, 2] → R

by x,

if x ∈ Q,

0,

if x ∉ Q.

f (x) = {

(7.2.14)

Show that f is not integrable on [1, 2].

Exercise 7.2.5 Suppose f is integrable on [a, b], and, for some real number m and M , m ≤ f (x) ≤ M for all x ∈ [a, b]. Show that b

m(b − a) ≤ ∫

f ≤ M (b − a).

(7.2.15)

a

7.2.1 Notation and Terminology The definition of the integral described in this section is due to Darboux. One may show it to be equivalent to the integral defined by Riemann. Hence functions that are integrable in the sense of this discussion are referred to as Riemann integrable functions and we call the integral the Riemann integral. This is in distinction to the Lebesgue integral, part of a more general theory of integration. We sometimes refer to this integral as the definite integral, as opposed to an indefinite integral, the latter being a name given to an antiderivative (a function whose derivative is equal to a given function). If f is integrable on [a, b], then we will also denote b

∫

f

(7.2.16)

f (x)dx.

(7.2.17)

a

by b

∫ a

The variable x in the latter is a "dummy" variable; we may just as well write b

∫

f (t)dt

(7.2.18)

a

7.2.2

https://math.libretexts.org/@go/page/22679

or b

∫

f (s)ds.

(7.2.19)

a

For example, if f

: [0, 1] → R

is defined by f (x) = x

2

,

then

1

∫ 0

1

f =∫

1 2

x dx = ∫

0

2

t dt.

(7.2.20)

0

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7.2.3

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7.3: Integrability Conditions Proposition 7.3.1 If a < b and f

: [a, b] → R

is monotonic, then f is integrable on [a, b].

Proof Suppose f is nondecreasing. Given ϵ > 0, let n ∈ Z

+

be large enough that

(f (b) − f (a))(b − a) < ϵ.

(7.3.1)

n

For i = 0, 1, … , n, let (b − a)i xi = a +

Let P

= { x0 , x1 , … , xn } .

.

(7.3.2)

n

Then n

U (f , P ) − L(f , P )

n

= ∑ f (xi ) (xi − xi−1 ) − ∑ f (xi−1 ) (xi − xi−1 ) i=1

i=1

n

b −a

= ∑ (f (xi ) − f (xi−1 )) i=1

n

b −a =

((f (x1 ) − f (x0 )) + (f (x2 ) − f (x1 )) + ⋯

n

+ (f (xn−1 ) − f (xn−2 )) + (f (xn ) − f (xn−1 ))) b −a =

(f (b) − f (a)) n

< ϵ.

Hence f is integrable on [a, b].

Q.E.D.

Example 7.3.1 Let φ : Q ∩ [0, 1] → Z

+

be a one-to-one correspondence. Define f

: [0, 1] → R

by

1 f (x) =

∑

φ(q)

q∈Q∩[0,1]

2

(7.3.3) .

q≤x

Then f is increasing on [0, 1], and hence integrable on [0, 1].

Proposition 7.3.2 If a < b and f

: [a, b] → R

is continuous, then f is integrable on [a, b].

Proof Given ϵ > 0, let ϵ γ =

.

(7.3.4)

b −a

Since f is uniformly continuous on [a, b], we may choose δ > 0 such that |f (x) − f (y)| < γ

whenever |x − y| < δ. Let P

= { x0 , x1 , … , xn }

(7.3.5)

be a partition with

7.3.1

https://math.libretexts.org/@go/page/22680

sup {| xi − xi−1 | : i = 1, 2, … , n} < δ.

(7.3.6)

mi = inf {f (x) : xi−1 ≤ x ≤ xi }

(7.3.7)

Mi = sup {f (x) : xi−1 ≤ x ≤ xi } ,

(7.3.8)

If, for i = 1, 2, … , n,

and

then M

i

− mi < γ.

Hence n

U (f , P ) − L(f , P )

n

= ∑ Mi (xi − xi−1 ) − ∑ mi (xi − xi−1 ) i=1

i=1

n

= ∑ (Mi − mi ) (xi − xi−1 ) i=1 n

< γ ∑ (xi − xi−1 ) i=1

= γ(b − a) = ϵ.

Thus f is integrable on [a, b].

Q.E.D.

Exercise 7.3.1 Suppose a < b, f [a, b].

: [a, b] → R

is bounded, and c ∈ [a, b]. Show that if

f

is continuous on

[a, b]∖{c},

then f is integrable on

Exercise 7.3.2 Suppose a < b and f is continuous on [a, b] with f (x) ≥ 0 for all x ∈ [a, b]. Show that if b

∫

f = 0,

(7.3.9)

a

then f (x) = 0 for all x ∈ [a, b].

Exercise 7.3.3 Suppose a < b and f is continuous on [a, b]. For i = 0, 1, … , n, n ∈ Z

+

let

,

(b − a)i xi = a +

and, for i = 1, 2, … , n, let c

i

∈ [ xi−1 , xi ] .

(7.3.10) n

Show that b

∫

n

b −a f = lim n

n→∞

a

∑ f (ci ) .

(7.3.11)

i=1

In the notation of Exercise 7.3.3, we call the approximation b

∫

b −a f ≈ n

a

a right-hand rule approximation if c

i

= xi ,

n

∑ f (ci )

a left-hand rule approximation if c

i

ci =

(7.3.12)

i=1

xi−1 + xi

.

= xi−1 ,

and a midpoint rule approximation if (7.3.13)

2

7.3.2

https://math.libretexts.org/@go/page/22680

These are basic ingredients in creating numerical approximations to integrals. This page titled 7.3: Integrability Conditions is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

7.3.3

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7.4: Properties of Integrals Proposition 7.4.1 If f

: D → R

and g : D → R, then sup{f (x) + g(x) : x ∈ D} ≤ sup{f (x) : x ∈ D} + sup{g(x) : x ∈ D}

(7.4.1)

inf{f (x) + g(x) : x ∈ D} ≥ inf{f (x) : x ∈ D} + inf{g(x) : x ∈ D}

(7.4.2)

and

Exercise 7.4.1 Prove the previous proposition.

Exercise 7.4.2 Find examples for which the inequalities in the previous proposition are strict.

Proposition 7.4.2 Suppose f and g are both integrable on [a, b]. Then f + g is integrable on [a, b] and b

∫

b

(f + g) = ∫

a

b

f +∫

a

g.

(7.4.3)

a

Proof Given ϵ > 0, let P and P be partitions of [a, b] with 1

2

ϵ U (f , P1 ) − L (f , P1 )

0,

let

and

P1

P2

be partitions of

[a, c]

and

[c, b],

ϵ U (f , P1 ) − L (f , P1 )

0 so that δ < ϵ and ϵ |f (x) − f (y)|

0 for all x ∈ [a, b], then there exists c ∈ [a, b] such that b

∫

b

f g = f (c) ∫

a

g.

(7.5.21)

a

Exercise 7.5.4 Prove the Generalized Integral Mean Value Theorem.

7.5.4

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7.5.5

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7.6: Taylor's Theorem Revisited The following is a version of Taylor's Theorem with an alternative form of the remainder term.

Theorem 7.6.1 (Taylor's Theorem) Suppose f

∈ C

(n+1)

(a, b), α ∈ (a, b),

and n

f

(k)

Pn (x) = ∑

(α)

k

(x − α ) .

(7.6.1)

k!

k=0

Then, for any x ∈ (a, b), x

f

(n+1)

f (x) = Pn (x) + ∫

(t)

n

(x − t) dt.

(7.6.2)

n!

α

Proof By the Fundamental Theorem of Calculus, we have x

∫

′

f (t)dt = f (x) − f (α),

(7.6.3)

α

which implies that x ′

f (x) = f (α) + ∫

f (t)dt.

(7.6.4)

α

Hence the theorem holds for n = 0. Suppose the result holds for n = k − 1, that is, x

f

(k)

(t)

f (x) = Pk−1 (x) + ∫

k−1

(x − t)

dt.

(7.6.5)

(k − 1)!

α

Let F (t) = f

(k)

(t),

(7.6.6)

k−1

(x − t) g(t) =

,

(7.6.7)

.

(7.6.8)

(k − 1)!

and k

(x − t) G(t) = − k!

Then x

f

(k)

(t)

∫ α

x k−1

(x − t)

dt = ∫

(k − 1)!

F (t)g(t)dt

α x ′

= F (x)G(x) − F (α)G(α) − ∫

F (t)G(t)dt

α

f

(k)

k

x

(α)(x − α )

=

f

(k+1)

+∫ k!

(t)

k

(x − t) dt, k!

α

Hence x

f

(k+1)

f (x) = Pk (x) + ∫ α

and so the theorem holds for n = k .

(t)

k

(x − t) dt,

(7.6.9)

k!

Q.E.D.

7.6.1

https://math.libretexts.org/@go/page/22683

Exercise 7.6.1 (Cauchy form of the remainder) Under the conditions of Taylor's Theorem as just stated, show that x

f

(n+1)

(t)

∫

f

n

(n+1)

(γ)

(x − t) dt = n!

α

n

(x − γ ) (x − α)

(7.6.10)

n!

for some γ between α and x.

Exercise 7.6.2 (Lagrange form of the remainder) Under the conditions of Taylor's Theorem as just stated, show that x

f

(n+1)

∫ α

for some γ between α and restrictive assumptions.

x.

(t)

n

f

(n+1)

(γ)

(x − t) dt = n!

n+1

(x − α )

(7.6.11)

(n + 1)!

Note that this is the form of the remainder in Theorem

6.6.1,

although under slightly more

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7.6.2

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7.7: An Improper Integral Definition If f integrable on [a, b] for all b > a and b

lim

∫

b→+∞

f (x)dx

(7.7.1)

a

exists, then we define +∞

∫

b

f (x)dx =

lim

∫

b→+∞

a

f (x)dx.

(7.7.2)

a

If f is integrable on [a, b] for all a < b and b

lim

∫

a→−∞

f (x)dx

(7.7.3)

a

exists, then we define b

b

∫

f (x)dx =

lim

∫

a→−∞

−∞

f (x)dx.

(7.7.4)

a

Both of these integrals are examples of improper integrals.

Proposition 7.7.1 Suppose f is continuous on [a, ∞) and f (x) ≥ 0 for all x ≥ a. If there exists g : [a, +∞) → R for which +∞

∫

g(x)dx

(7.7.5)

f (x)dx

(7.7.6)

a

exists and g(x) ≥ f (x) for all x ≥ a , then +∞

∫ a

exists.

Exercise 7.7.1 Prove the preceding proposition.

Example 7.7.1 Suppose 1 f (x) =

(7.7.7)

2

1 +x

and 1, g(x) = {

if 0 ≤ x < 1, (7.7.8)

1 2

x

,

if x ≥ 1.

Then, for b > 1 b

∫ 0

1

g(x)dx = ∫ 0

b

dx + ∫ 1

1 2

x

7.7.1

1 dx = 1 + 1 −

1 =2−

b

,

(7.7.9)

b

https://math.libretexts.org/@go/page/25413

so +∞

∫

1 g(x)dx = lim (2 −

0

) = 2.

(7.7.10)

b

b→∞

Since 0 < f (x) ≤ g(x) for all x ≥ 0 , it follows that +∞

∫

1 dx

(7.7.11)

dx < 2.

(7.7.12)

1 + x2

0

exists, and, moreover, +∞

1

∫

2

1 +x

0

Also, the substitution u = −x shows that 0

∫ −∞

0

1 2

1 +x

+∞

+∞

1

dx = − ∫

2

1 +u

du = ∫ 0

1 2

du.

(7.7.13)

1 +u

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7.7.2

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CHAPTER OVERVIEW 8: More Functions 8.1: The Arctangent Function 8.2: The Tangent Function 8.3: The Sine and Cosine Functions 8.4: The Logarithm Functions 8.5: The Exponential Function Thumbnail: Periodic functions. (CC BY-NC-SA; OpenStax) This page titled 8: More Functions is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1

8.1: The Arctangent Function Definition For any x ∈ R, we call x

1

arctan(x) = ∫

2

dt

(8.1.1)

1 +t

0

the arctangent of x.

Proposition 8.1.1 The arctangent function is differentiable at every x ∈ R. Moreover, if f (x) = arctan(x), then 1

′

f (x) =

.

2

(8.1.2)

1 +x

Proof The result follows immediately from Theorem 7.5.4.

Q.E.D.

Proposition 8.1.2 The arctangent is increasing on R. Proof The result follows immediately from the previous proposition and the fact that 1 2

>0

(8.1.3)

1 +x

for every x ∈ R .

Q.E.D.

Definition +∞

π = 2 limx→+∞ arctan(x) = 2 ∫

0

1 2

1+t

dt

Note that 0 < π < 4 by Example 7.7.1. The following proposition says that the arctangent function is an odd function.

Theorem 8.1.3 For any x ∈ R, arctan (x) = − arctan(−x). Proof Using the substitution t = −u, we have x

−x

1

arctan(x) = ∫ 0

1

dt = − ∫ 2

1 +t

0

du = − arctan(−x). 1 + u2

Q.E.D. It now follows that

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π lim

arctan(x) = −

x→−∞

lim

arctan(−x) = −

x→−∞

Hence the range of the arctangent function is (−

π 2

,

π 2

.

(8.1.4)

2

.

)

Proposition 8.1.4 If x > 0, then 1

π

arctan(x) + arctan(

) =

.

x

(8.1.5)

2

Proof Using the substitution t =

1 u

,

we have 1

1 arctan(

1

x

) =∫ x

2

dt

1 +t

0 x

1

=∫ 1+

+∞

1 (−

1

2

) du

u

2

u x

1

= −∫

du 2

1 +u

+∞ +∞

1

=∫

2

x

π =

du

1 +u

x

−∫ 2

1 2

du

1 +u

0

π =

− arctan(x). 2

Q.E.D.

Proposition 8.1.5 If x < 0 , then 1 arctan(x) + arctan(

π ) =−

x

.

(8.1.6)

2

Proof The result follows immediately from the preceding proposition and the fact that arctangent is an odd function.

Exercise 8.1.1 Show that arctan (1) =

π 4

and arctan(−1) = − . π 4

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8.1.2

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8.2: The Tangent Function Let π A ={

+ nπ : n ∈ Z}

(8.2.1)

2

and D = R∖A . Let π t : (−

π ,

2

) → R

(8.2.2)

2

be the inverse of the arctangent function. Note that t is increasing and differentiable on (− on D as follows: For any x ∈ D, let

π 2

,

π 2

).

We may extend

t

to a function

π g(x) = sup {n : n ∈ Z, −

+ nπ < x}

(8.2.3)

2

and define T (x) = t(x − g(x)π) .

Definition With the notation of the above discussion, for any x ∈ D, we call the value T (x) the tangent of x, which we denote tan(x).

Proposition 8.2.1 The tangent function has domain D (as defined above), range tangent function is increasing on each interval of the form π (−

and is differentiable at every point

x ∈ D.

Moreover, the

π + nπ,

2 n ∈ Z,

R,

+ nπ) ,

(8.2.4)

2

with π tan((

+ nπ) +) = −∞

(8.2.5)

+ nπ) −) = +∞.

(8.2.6)

2

and π tan(( 2

Proof These results follow immediately from our definitions.

Q.E.D.

Definition Let E ⊂ R . We say a function f : E → R is periodic if there exists a real number p > 0 such that, for each x ∈ E, x + p ∈ E and f (x + p) = f (x). We say p is the period of a periodic function f if p is the smallest positive number for which f (x + p) = f (x) for all x ∈ E.

Proposition 8.2.2 The tangent function has period π. Proof The result follows immediately from our definitions.

Q.E.D.

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Proposition 8.2.3 (Addition formula for tangent) For any x, y ∈ D with x + y ∈ D , tan(x) + tan(y) tan(x + y) =

.

(8.2.7)

1 − tan(x) tan(y)

Proof Suppose

y1 , y2 ∈ (−

x1 x2 ≥ 1

π 2

,

π 2

)

with

y1 + y2 ∈ (−

would imply that

π 2

,

π 2

Let

).

x1 = tan(y1 )

and

x2 = tan(y2 ).

Note that if

x1 > 0,

then

1 x2 ≥

,

(8.2.8)

x1

which in turn implies that y1 + y2 = arctan(x1 ) + arctan(x2 ) 1 ≥ arctan(x1 ) + arctan(

) x1

π =

, 2

contrary to our assumptions. Similarly, if x

1

< 0,

then x

1 x2

≥1

would imply that

1 x2 ≤

,

(8.2.9)

x1

which in turn implies that y1 + y2 = arctan(x1 ) + arctan(x2 ) 1 ≤ arctan(x1 ) + arctan(

) x1

π =−

, 2

contrary to our assumptions. Thus we must have x > 0, then

x1 x2 < 1.

Moreover, suppose

u

is a number between

−x1

and

x2 .

If

1

1 x2

.

(8.2.13)

x1

Now let x =

x1 + x2

.

(8.2.14)

1 − x1 x2

We want to show that

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arctan(x) = arctan(x1 ) + arctan(x2 ),

(8.2.15)

which will imply that tan(y1 ) + tan(y2 ) = tan(y1 + y2 ).

(8.2.16)

1 − tan(y1 ) tan(y2 )

We need to compute x

arctan(x) = arctan(

x1 + x2

1

+x

1−x

) =∫

1 − x1 x2

1

2

x

1

2

dt.

(8.2.17)

2

1 +t

0

Let x1 + u

t = φ(u) =

,

(8.2.18)

1 − x1 u

where u varies between −x

1,

where t = 0, and x , where t = x. Now 2

2

1 +x

(1 − x1 u) − (x1 + u) (−x1 )

′

φ (u) =

1

= 2

(1 − x1 u)

(1 − x1 u)

,

(8.2.19)

2

which is always positive, thus showing that φ is an increasing function, and 1

1 =

1 + t2

2

x1 +u

1 +(

1−x1 u

)

(1 − x1 u) = (1 − x1 u)

2

2

+ (x1 + u)

(1 − x1 u)

2

2

=

. 2

2

(1 + x ) (1 + u ) 1

Hence x2

1

arctan(x) = ∫ −x1 0

x2

1

=∫ −x1

du

2

1 +u

2

1 +u −x1

1

du + ∫ 0

2

du

1 +u

1

= −∫ 1 + u2

0

du + arctan(x2 )

= − arctan(−x1 ) + arctan(x2 ) = arctan(x1 ) + arctan(x2 ).

Now suppose y

1,

y2 ∈ (−

π 2

,

π 2

)

with y

1

+ y2 >

π 2

.

Then y

1

+ y2 ∈ (

π 2

,

, π) x1 > 0, x2 > 0,

and

1 x2 >

.

(8.2.20)

x1

With u and x as above, note then that as u increases from −x to 1

1 x1

to x

2,

t

1 x1

,t

increases from 0 to +∞, and as u increases from

increases from −∞ to x.

Hence we have

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x

arctan(x) + π

0

1

=∫

2

1 +t

0 x

+∞

1

dt + ∫

2

1 +t

−∞

+∞

1

=∫

dt + ∫ 0

−∞ x2

1 +t

x

x2

−x1

dt 1 x

2

1

1

du + ∫

1 +u

−x1

1

=∫

dt

1 + t2

0

1

=∫ 1

2

1 +t

1

dt + ∫ 2

1

2

du

1 +u

1 2

du

1 +u

= arctan(x2 ) − arctan(−x1 ) = arctan(x2 ) + arctan(x1 ).

Hence tan(y1 + y2 ) = tan(y1 + y2 − π) = tan(arctan(x)) =

x1 + x2 1 − x1 x2 tan(y1 ) + tan(y2 )

=

. 1 − tan(y1 ) tan(y2 )

The case when x < 0 may be handled similarly; it then follows that the addition formula holds for all y , y ∈ (− , ) . The case for arbitrary y , y ∈ D with y + y ∈ D then follows from the periodicity of the tangent function. Q.E.D. 1

1

1

2

1

2

π

π

2

2

2

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8.2.4

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8.3: The Sine and Cosine Functions We begin by defining functions π s : (−

π ,

2

] → R

(8.3.1)

] → R

(8.3.2)

2

and π c : (−

π ,

2

2

by tan(x)

⎧ ⎪ s(x) = ⎨ ⎩ ⎪

,

if x ∈ (−

π 2

2

√1+tan (x)

π

,

),

2

(8.3.3)

1,

if x =

π 2

and 1

⎧ ⎪ c(x) = ⎨ ⎩ ⎪

,

π

if x ∈ (−

2

2

√1+tan (x)

π

,

),

2

(8.3.4)

0,

if x =

π 2

.

Note that lim x→

π

−π

s(x) = −

lim y→+∞

2

y − −−− − = √1 + y2

1 −−−− − =1

lim y→+∞

√1 +

(8.3.5)

1

y

2

and 1

1 lim x→

π 2

c(x) =

lim y→+∞

−

y

− −−− − 2 √1 + y

=

lim y→+∞

−−−− −

= 0,

(8.3.6)

1

√1 +

y2

which shows that both s and c are continuous functions. Next, we extend the definitions of s and c to functions π S : (−

3π ,

2

] → R

(8.3.7)

] → R

(8.3.8)

2

and π C : (−

3π ,

2

2

by defining s(x),

π

if x ∈ (−

S(x) = {

π

−s(x − π),

if x ∈ (

c(x),

if x ∈ (−

2

,

2

π 2

3π

,

], (8.3.9)

]

2

and C (x) = { −c(x − π),

if x ∈ (

π 2

π 2

,

,

π 2

3π 2

], (8.3.10)

].

Note that lim x→

π 2

+

S(x) =

lim x→−

π 2

−s(x) = − +

lim y→−∞

y 1 − −−− − = − lim −−−− − =1 2 y→−∞ 1 √1 + y −√ 1 + 2

(8.3.11)

y

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and 1

1 lim x→

π 2

C (x) =

+

lim x→−

π 2

−c(x) = −

which shows that both S and C are continuous at

lim y→−∞

+

π 2

.

y

=−

− −−− − √1 + y2

lim y→−∞

−−−− − −√ 1 +

= 0,

(8.3.12)

1

y

2

Thus both S and C are continuous.

Finally, for any x ∈ R, let π g(x) = sup {n : n ∈ Z, −

+ 2nπ < x}

(8.3.13)

2

and define sin(x) = S(x − 2πg(x))

(8.3.14)

cos(x) = C (x − 2πg(x)).

(8.3.15)

and

Definition With the notation as above, for any x ∈ R, we call sin(x) and cos(x) the sine and cosine of x, respectively.

Proposition 8.3.1 The sine and cosine functions are continuous on R. Proof From the definitions, it is sufficient to verify continuity at

3π 2

.

Now 3π

lim x→

3π 2

sin(x) =

−

lim x→

3π 2

S(x) = S (

π ) = −s (

2

−

) = −1

(8.3.16)

2

and lim x→

3π 2

sin(x) =

+

lim x→

=

3π 2

S(x − 2π)

+

lim x→−

π 2

s(x) +

y =

lim y→−∞

− −−− − 2 √1 + y 1

=

lim y→−∞

−−−− − −√ 1 +

1

y

2

= −1,

and so sine is continuous at

3π 2

. Similarly, 3π lim x→

3π 2

−

cos(x) =

lim x→

3π 2

C (x) = C (

−

π ) = −c (

2

) =0

(8.3.17)

2

and

8.3.2

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lim x→

3π 2

cos(x) =

+

lim x→

=

3π

C (x − 2π)

+

2

lim x→−

π 2

c(x) +

1 =

lim y→−∞

− −−− − √1 + y2 1 y

=

lim y→−∞

−−−− − −√ 1 +

1

y

2

= 0,

and so cosine is continuous at

3π 2

.

Q.E.D.

8.3.1 Properties of sine and cosine Proposition 8.3.2 The sine and cosine functions are periodic with period 2π. Proof The result follows immediately from the definitions.

Q.E.D.

Proposition 8.3.3 For any x ∈ R, sin(−x) = − sin(x) and cos(−x) = cos(x). Proof The result follows immediately from the definitions.

Q.E.D.

Proposition 8.3.4 For any x ∈ R, sin

2

2

(x) + cos (x) = 1

.

Proof The result follows immediately from the definition of s and c.

Q.E.D.

Proposition 8.3.5 The range of both the sine and cosine functions is [−1, 1]. Proof The result follows immediately from the definitions along with the facts that − −−− − √1 + y

2

− − ≥ √y

2

= |y|

(8.3.18)

and − −−− − √1 + y

for any y ∈ R.

2

≥1

(8.3.19)

Q.E.D

8.3.3

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Proposition 8.3.6 For any x in the domain of the tangent function, sin(x) tan(x) =

.

(8.3.20)

cos(x)

Proof The result follows immediately from the definitions.

Q.E.D.

Proposition 8.3.7 For any x in the domain of the tangent function, 2

tan (x)

2

sin (x) =

(8.3.21)

2

1 + tan (x)

and 1

2

cos (x) =

.

2

(8.3.22)

1 + tan (x)

Proof The result follows immediately from the definitions.

Q.E.D.

Proposition 8.3.8 For any x, y ∈ R, cos(x + y) = cos(x) cos(y) − sin(x) sin(y).

(8.3.23)

Proof First suppose x, y, and x + y are in the domain of the tangent function. Then 1

2

cos (x + y) =

2

1 + tan (x + y) 1 =

2

tan(x)+tan(y)

1 +(

1−tan(x) tan(y)

) 2

(1 − tan(x) tan(y)) =

2

1 − tan(x) tan(y))

2

+ (tan(x) + tan(y)) 2

(1 − tan(x) tan(y)) = 2

2

(1 + tan (x))(1 + tan (y)) 2

⎛ = ⎜

− −−−−−−− − 2

⎞

tan(x) tan(y)

1 − −−−−−−− −

−

2

⎝ √ 1 + tan (x) √ 1 + tan (y)

− −−−−−−− − 2

− −−−−−−− −⎟ 2

√ 1 + tan (x) √ 1 + tan (y) ⎠

2

= (cos(x) cos(y) − sin(x) sin(y)) .

Hence cos(x + y) = ±(cos(x) cos(y) − sin(x) sin(y)).

(8.3.24)

Consider a fixed value of x . Note that the positive sign must be chosen when y = 0. Moreover, increasing y by π changes the sign on both sides, so the positive sign must be chosen when y is any multiple of π . Since sine and cosine are

8.3.4

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continuous functions, the choice of sign could change only at points at which both sides are separated by a distance of π, so we must always choose the positive sign. Hence we have cos(x + y) = cos(x) cos(y) − sin(x) sin(y)

0,

but these points are

(8.3.25)

for all x, y ∈ R for which x, y, and x + y are in the domain of the tangent function. The identity for the other values of x and y now follows from the continuity of the sine and cosine functions. Q.E.D.

Proposition 8.3.9 For any x, y ∈ R, sin(x + y) = sin(x) cos(y) + sin(y) cos(x).

(8.3.26)

Exercise 8.3.1 Prove the previous proposition.

Exercise 8.3.2 Show that for any x ∈ R, π sin(

− x) = cos(x)

(8.3.27)

− x) = sin(x).

(8.3.28)

2

and π cos( 2

Exercise 8.3.3 Show that for any x ∈ R, sin(2x) = 2 sin(x) cos(x)

(8.3.29)

and 2

2

cos(2x) = cos (x) − sin (x).

(8.3.30)

Exercise 8.3.4 Show that for any x ∈ R, 1 − cos(2x)

2

sin (x) =

(8.3.31) 2

and 1 + cos(2x)

2

cos (x) =

.

(8.3.32)

2

Exercise 8.3.5 Show that π sin(

π ) = cos(

4

4

π sin(

1 ) =

–, √2

π ) = cos(

6

8.3.5

1 ) =

3

(8.3.33)

,

(8.3.34)

2

https://math.libretexts.org/@go/page/22687

and π sin(

– √3

π ) = cos(

3

) =

.

6

(8.3.35)

2

8.3.2 The calculus of the trigonometric functions Proposition 8.3.10 limx→0

arctan(x) x

=1

.

Proof Using l'Hôpital's rule, 1

arctan(x) lim

1+x2

= lim x

x→0

= 1.

(8.3.36)

1

x→0

Q.E.D

Proposition 8.3.11 tan(x)

limx→0

=1

x

.

Proof Letting x = arctan(u), we have tan(x) lim

u = lim

x

x→0

u→0

= 1.

(8.3.37)

arctan(u)

Q.E.D

Proposition 8.3.12 sin(x)

limx→0

=1

x

.

Proof We have sin(x) lim x→0

tan(x) = lim

x

x→0

cos(x) = 1.

(8.3.38)

x

Proposition 8.3.13 1−cos(x)

limx→0

x

= 0.

Proof We have

8.3.6

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1 − cos(x) lim

1 − cos(x) = lim (

x

x→0

1 + cos(x) )(

x

x→0

) 1 + cos(x)

2

1 − cos (x) = lim x→0

x(1 + cos(x)) sin(x)

= lim (

sin(x) )(

x

x→0

) 1 + cos(x)

= (1)(0) = 0.

Q.E.D

Proposition 8.3.14 If f (x) = sin(x), then f

′

(x) = cos(x).

Proof We have sin(x + h) − sin(x)

′

f (x) = lim h

h→0

sin(x) cos(h) + sin(h) cos(x) − sin(x) = lim h

h→0

cos(h) − 1 = sin(x) lim h→0

sin(h) + cos(x) lim

h

h→0

h

= cos(x).

Q.E.D

Proposition 8.3.15 If f (x) = cos(x), then f

′

(x) = − sin(x)

.

Exercise 8.3.6 Prove the previous proposition.

Definition For appropriate x ∈ R, we call cos x cot(x) =

,

(8.3.39)

,

(8.3.40)

sin(x) 1 sec(x) = cos(x)

and 1 csc(x) =

(8.3.41) sin(x)

the cotangent, secant, and cosecant of x, respectively.

8.3.7

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Exercise 8.3.7 If f (x) = tan(x) and g(x) = cot(x), show that ′

2

f (x) = sec (x)

(8.3.42)

and ′

2

g (x) = − csc (x).

(8.3.43)

f (x) = sec(x) tan(x),

(8.3.44)

Exercise 8.3.8 If f (x) = sec(x) and g(x) = csc(x), show that ′

and ′

g (x) = − csc(x) cot(x).

(8.3.45)

Proposition 8.3.16 2∫

1

−1

− −−− − √1 − x2 dx = π

.

Proof Let x = sin(u) . Then as u varies from −

π 2

to

π 2

,x

varies from −1 to 1. And, for these values, we have

− − − − − − − − −

− − − − − − − −−− − 2 2 2 √ 1 − x = √ 1 − sin (u) = √cos (u) = | cos(u)| = cos(u).

(8.3.46)

Hence 1

∫

π

− −−− − √ 1 − x2 dx = ∫

−1

2

2

cos (u)du −

π 2 1 2

1 + cos(2u)

π

2

=∫

du −

2 1 2

π

1

=∫

1 du +

−

π

2

2

2

π

2

∫

cos(2u)du −

π 2

1

=

+ 2

(sin(π) − sin(−π)) 4

π =

. 2

Q.E.D.

Exercise 8.3.9 Find the Taylor polynomial P of order 9 for f (x) = sin(x) at 0. Note that this is equal to the Taylor polynomial of order 10 for f at 0. Is P ( ) an overestimate or an underestimate for sin ( )? Find an upper bound for the error in this approximation. 9

9

1

1

2

2

This page titled 8.3: The Sine and Cosine Functions is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

8.3.8

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8.4: The Logarithm Functions Definition Given a positive real number x, we call x

1

log(x) = ∫

dt

(8.4.1)

t

1

the logarithm of x. Note that log(1) = 0, log(x) < 0 when 0 < x < 1, and log(x) > 0 when x > 1.

Proposition 8.4.1 The function f (x) = log(x) is an increasing, differentiable function with 1

′

f (x) =

(8.4.2) x

for all x > 0. Proof Using the Fundamental Theorem of Calculus, we have 1

′

f (x) =

>0

(8.4.3)

x

for all x > 0, from which the result follows.

Q.E.D.

Proposition 8.4.2 For any x > 0 , 1 log(

) = − log(x).

(8.4.4)

x

Proof Using the substitution t =

1 u

,

we have 1

1 log(

x

x

1

x

1

) =∫

dt = ∫ t

1

x

1 u (−

1

) du = − ∫ 2

u

1

du = − log(x).

(8.4.5)

u

Q.E.D.

Proposition 8.4.3 For any positive real numbers x and y, log(xy) = log(x) + log(y).

(8.4.6)

Proof Using the substitution t = xu, we have

8.4.1

https://math.libretexts.org/@go/page/22688

xy

1

log(xy) = ∫

dt t

1 y

x

=∫

du xu

1 x

1

y

1

=∫

1

du + ∫ u

1

1

x

du u

1 x

1

= −∫

du + log(y) u

1

1 = − log(

) + log(y) x

= log(x) + log(y).

Q.E.D.

Proposition 8.4.4 If r ∈ Q and x is a positive real number, then r

log(x ) = r log(x).

(8.4.7)

Proof Using the substitution t = u

r

we have

,

r

x r

1

x

1

log(x ) = ∫

r−1

t

x

ru

dt = ∫

r

u

1

1

du = r ∫ 1

du = r log(x).

(8.4.8)

u

Q.E.D.

Proposition 8.4.5 limx→+∞ log(x) = +∞

and lim

x→0+

log(x) = −∞.

Proof Given a real number M , choose an integer n for which n log(2) > M (there exists such an n since log(2) > 0 ). Then for any x > 2 , we have n

n

log(x) > log(2 ) = n log(2) > M .

Hence lim

x→+∞

log(x) = +∞

(8.4.9)

.

Similarly, given any real number M , we may choose an integer we have

n

for which

−n log(2) < M .

Then for any 0 < x

0 , lim

α

x

= +∞.

(8.4.11)

x→+∞

8.4.2

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Proposition 8.4.6 For any rational number α > 0 , log(x) lim

= 0.

α

(8.4.12)

x

x→+∞

Proof Choose a rational number β such that 0 < β < α. Now for any t > 1 , 1

1